**CURRICULUM VITAE**

**JAMES A.
YORKE**

**December 2011**

Chair of Mathematics

Distinguished
University Professor of Mathematics and

Web page: http://yorke.umd.edu

e-mail: yorke at umd.edu

phone: 301-405-5051

Born
1941 in

**Education**

University of

**Professional Positions**

Appointments in the IPST (“IPST” denotes the Institute for Physical Science and Technology or in its predecessors, at the University of Maryland). JY has held joint appointments with Mathematics since 1976 and Physics since 2000.

Research Associate, 1966-1967

Research Assistant Professor
1967-1969

Research Associate Professor
1969-1973

Professor 1973-present

Director of IPST (Acting
Director 1985-1988) 1988-Dec 2001

Distinguished University
Professor since 1995

Director
of Applied Math. and Scientific Computing (

graduate program 2006-2007

Chair of Mathematics, 2007- (a five year appointment)

Expert (part-time appointment) National Cancer Institute 1978-1979

**Honors and Awards**

Jurgen Moser lecture/award 2011 presented in Snowbird Utah

Norbert Wiener Lecturer – Tufts University Fall 2006

Commencement address May 2006 University School of Health Information Sciences

Penn State University 2006 Marker Lecturer in Mathematics

Fellow of the American Physical Society, appointed 2003

Japan Prize Laureate 2003 (shared with Benoit Mandelbrot); see www.japanprize.jp The Japan Prize for Science and Technology is a Japanese version of the Nobel Prize. One is awarded in medical science and one in the rest of science and technology. The Emperor of Japan presides over the awards ceremony.

Distinguished Alumnus Award 2002, alumnus of U of Md. College of Computer Math, and Physics Sciences.

An

AAAS Fellow - elected 1998

First UMCP
recipient of the

38^{th} Annual Chaim
Weizmann Memorial Lecturer

- Weizmann Institute Rehovot,

Distinguished University Professor - appointed 1995** **

Guggenheim fellow 1980

** **

**Principal investigator on current research
grants **

NSF
grants 2001-2010 Applications of Nonlinear Dynamics

NIH grant 2003-10 Reliable Assembler for Whole Genome Shotgun Data

**Editorial
boards**

International Journal of
Bifurcation and Chaos

Journal of Difference Equations and Applications

** **

**Current Research Projects**

See: http://yorke.umd.edu/current-projects.html

**Publications
**

**A.
Books: **

1984 H. W. Hethcote and J. A. Yorke,

*Gonorrhea
Transmission Dynamics and Control*, Springer-Verlag Lecture Notes in
Biomathematics #56, 1984.

1994 E. Ott, T. Sauer and J. A. Yorke,

*Coping
with Chaos*, 1994 John Wiley & Sons, Inc.

1997 K. Alligood, T. Sauer and J. A. Yorke,

*Chaos:
An Introduction to Dynamical Systems*,

1997 H. E. Nusse and J. A. Yorke,

*Dynamics:
Numerical Explorations*, Applied Mathematical Sciences 101,

1997
C. Grebogi and. J. A. Yorke, Editors*,*

* The Impact of Chaos on Science and Society*,
United Nations University Press, Tokyo (1997). ISBN 92-808-0882-6.

**B. Journal Papers **

**1967**

1. A. Strauss and J. A. Yorke, Perturbation theorems for ordinary differential equations, J. Differential Equations 1 (1967), 15-30.

2. J. A. Yorke, Invariance for ordinary differential equations. Math. Systems Theory 1 (1967), 353‑372.

3. A. Strauss and J. A. Yorke, On asymptotically autonomous differential equations. Math. Systems Theory 1 (1967), 175-182.

**1968**

1. A. Strauss and J. A. Yorke, Perturbing asymptotically stable differential equations, Bull. Amer. Math. Soc. 74 (1968), 992-996. Announcement of #1969-7.

2. J. A. Yorke, Extending Lyapunov's second method to non-Lipschitz Lyapunov functions, Bull. Amer. Math. Soc. 74 (1968), 322-325. Announcement of #1970-3.

**1969 **

1. J. A. Yorke, Permutations and two sequences with the same cluster set, Proc. Amer. Math. Soc. 20 (1969), 606.

2. Elliot Winston and J. A. Yorke, Linear delay differential equations whose solutions become identically zero, Rev. Roumaine Math. Pures Appl. 14 (1969), 885-887.

Abstract: Linear delay differential equations with the property that all solutions become identically zero after a finite period of time are discussed.

3. A. Strauss and J. A. Yorke, Identifying perturbations which preserve asymptotic stability, Proc. Amer. Math. Soc. 22 (1969), 513-518.

4. N. P. Bhatia, G. P. Szego and J. A. Yorke, A Lyapunov characterization of attractors, Boll. Un. Mat. Ital. 4 (1969), 222-228.

Abstract: Necessary and sufficient conditions for a compact set to be respectively a global weak attractor and global attractor for the dynamical system defined by an ordinary differential equation are proved. These conditions are given by means of lower-semicontinuous Liapunov functions.

5. G. S. Jones and J. A. Yorke, The existence and nonexistence of critical points in bounded flows, J. Differential Equations 6 (1969), 238-247.

6. A. Strauss and J. A. Yorke,
On the fundamental theory of differential equations,

7. A. Strauss and J. A. Yorke, Perturbing uniform asymptotically stable non-linear systems, J. Differential Equations 6 (1969), 452-483. Announcement in #1968-1.

8. A. Strauss and J. A. Yorke, Perturbing uniformly stable linear systems with and without attraction, SIAM J. Appl. Math. 17 (1969), 725-739.

9. J. A. Yorke, Non-continuable solutions of differential-delay equations, Proc. Amer. Math. Soc. 21 (1969), 648-652.

Note. This paper discusses differential delay equations x’ = G(xT) with continuous G but with highly non-unique solutions of initial value problems. As a side issue, this paper contains a short proof of the Tietze Extension Theorem on metric spaces. If g is continuous on a closed set S in a metric space X, then define G = g on S and for x not in S,

G(x) = inf for y in S of {g(y) + d(x,y)/d(x,S) – 1}. Then G is continuous on X.

The Tietze extension proof is for functions with a lower bound.

10. J. A. Yorke, Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc. 22 (1969), 509-512.

Abstract. Assume dx/dt = F(x) is a differential equation on Rn or on a Hilbert space. Assume F satisfies the Lipshitz condition

|| F(x) – F(y) || ≤ L || x – y || where || . || denotes the Euclidean metric.

Assume p is a periodic orbit with period T. Then T ≥ 2 pi / L.

**1970 **

1. J. A. Yorke, Asymptotic stability for one-dimensional differential delay-equations, J. Differential Equations 7 (1970), 189-202.

2. J. A. Yorke, A continuous differential equation in Hilbert space without existence, Funkcialaj Ekvacioj 13 (1970), 19-21.

3. J. A. Yorke, Differential inequalities and non-Lipschitz scalar functions, Math. Systems Theory 4 (1970), 140-153.

4. Gerald S. Goodman and J. A. Yorke, Misbehavior of solutions of the differential equation dy/dx = f(x,y) + epsilon, when the right side is discontinuous, Mathematica Scandinavica 27 (1970), 72-76.

Abstract: It is well known that by consideration of the corresponding integral equation, most qualitative theorems concerning initial-value problems for the first order ordinary differential equation dy/dx = f (x,y) can be extended to the case where the right side is no longer continuous. In this note, however, we shall show by example that more than one widely used theorem in the continuous case cannot be so extended, at least not in a form that would preserve its most useful feature, as soon as the right side of the equation fails to be jointly continuous at just a single point, even though it remains bounded and continuous there in each variable separately.

5. A. Strauss and J. A. Yorke, Linear perturbations of ordinary differential equations , Proc. Amer. Math. Soc. 26 (1970), 255-260.

Abstract: We present several results dealing with the problem of the preservation of the stability of a system dx/dt=A(t)x that is subject to linear perturbations B(t)x, or to perturbations dominated by linear ones.

6. J. A. Yorke, A theorem on Lyapunov functions using the second derivative of V, Math. Systems Theory 4 (1970), 40-45.

**1971 **

1. A. Lasota and J. A. Yorke, Oscillatory solutions of a second order ordinary differential Equation, Ann. Polon. Math. 25 (1971), 175-178.

2. J. A. Yorke, Another proof of the Lyapunov convexity theorem, SIAM J. Control (1971), 9 351-353.

Abstract: A new proof of the Liapunov convexity theorem is presented.

3. S. Saperstone and J. A. Yorke, Controllability of linear oscillatory systems using positive controls, SIAM J. Control 9 (1971), 253-272.

Abstract: A linear autonomous control process is considered where the null control is an extreme point of the restraint set S. In the even that S=[0,1] (hence, scalar control) necessary and sufficient conditions are given so that the reachable set from the origin (in phase space) contains the origin as an interior point. For vector-valued controls with each component in [0,1], sufficient conditions are given so that the reachable set from the origin of a nonlinear autonomous control process contains the origin as an interior point.

4. A. Lasota and J. A. Yorke, Bounds for periodic solutions of differential equations in Banach spaces, J. Differential Equations 10 (1971), 83-91.

**1972 **

1. A. Lasota and J. A. Yorke, Existence of solutions of two-point boundary value problems for nonlinear systems, J. Differential Equations 11 (1972), 509-518.

2. J. A. Yorke, The maximum principle and controllability of nonlinear equations, SIAM J. Control 10 (1972), 334-338.

Abstract: The main result proved is that a nonlinear control equation is controllable if a related linear equation is controllable. The result allows the set of control values to be discrete and it is not assumed that small values of the control are available. The methods used are closely related to the Pontryagin maximum principle.

3. S. Grossman and J. A. Yorke, Asymptotic behavior and stability criteria for differential delay equations), J. Differential Equations 12 (1972), 236-255.

4. S. Bernfeld and J. A. Yorke, The behavior of oscillatory solutions of x"(t)+p(t)g(x(t))=0, SIAM J. Math. Anal. 3 (1972), 654-667.

Abstract: Various quantitative properties of oscillatory solutions of the scalar second order nonlinear differential equation are obtained under appropriate hypotheses on p and g. In particular, letting {ti, 0 < ti < ti+1, where ti goes to infinity, be the zeroes of any solution x(t), we obtain inequalities that yield asymptotic behavior on x(t). For example, it is shown that the integral of g(x(ti)) exists and is finite: moreover, assuming an added growth condition on g(x)/x, we have then that the integral of x(t) from 0 to infinity exists and is finite.

**1973 **

1. F. W. Wilson, Jr. and J. A. Yorke, Lyapunov functions and isolating blocks, J. Differential Equations 13 (1973), 106-123.

2. K. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci. 16 (1973), 75-101.

Abstract: At the present time VD is a major national problem. Essentially we are confronted with several epidemics. This paper is devoted to a study of processes of this nature. It is hoped that understanding of the mathematical nature of these processes will help in their control.

3,4. W. London, M.D. and J. A. Yorke, Recurrent outbreaks of measles, chicken pox, and mumps, I. Seasonal variation in contact rates, and II. Systematic differences in contact rates and stochastic effects, Amer. J. Epidemiology 98 (1973), 453-468 and 469-482.

5. A. Lasota and J. A. Yorke, The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations 13 (1973), 1-12.

6. A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488.

Abstract: A class of piecewise
continuous, piecewise C^{1} transformations on the interval [0,1] is
shown to have absolutely continuous invariant measures. This is the first paper
to show the existence of invariant measures defined on part of a space by taking
Lebesgue measure on the whole space and pushing it forward. This result shows
the existence of invariant measures for maps such as the tent map with slope s
where 1 < s ≤ 1. Such measures
were later called SRB measures when the limit measure is unique. This paper
also shows that if the map has slope 1 at one point, there need be no invariant
measure.

7. J. A. Yorke and W. N. Anderson, Predator-prey patterns, Proc. Nat. Acad. Sci. 70 (1973), 2069-2071.

Abstract: A graph-theoretic condition is given for the existence of stable solutions to the Volterra-Lotka equations.

**1974 **

1. S. N. Chow and J. A. Yorke, Lyapunov theory and perturbations of stable and asymptotically stable systems, J. Differential Equations 15 (974), 308-321.

2. J. L. Kaplan and J. A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay questions, J. Math. Anal. Appl. 48 (1974), 317-324.

3. J. L. Kaplan, A. Lasota and J. A. Yorke, An application of the Wazewski retract method to boundary value problems, Zeszyty Nauk. Uniw. Jagiellon 356 (1974), 7-14.

**1975 **

1. J. L. Kaplan and J. A. Yorke, On the stability of a periodic solution of a differential equation, SIAM J. Math. Anal. 6 (1975), 268-282.

Abstract: This paper considers the class of scalar, first order, differential delay equations y'(t) = -f(y(t-1)). It is shown that under certain restrictions there exists an annulus A in the (y(t), y(t-1)) - plane whose boundary is a pair of slowly oscillating periodic orbits and A is asymptotically stable. These results are applied to the frequently studied equation dx/dt = -ax(t-1)[1+ x(t)]. The techniques used are related to the Poincare-Bendixson method, used in the (y(t), y(t-1)) - plane.

2. T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992.

**1976 **

1. J. C. Alexander and J. A. Yorke, The implicit function theorem and the global methods of cohomology, J. Functional Anal. 21 (1976), 330-339.

2. A. Lajmanovich Gergely and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), 221-236.

Abstract: The spread of gonorrhea in a population is highly nonuniform. The mathematical model discussed takes this into account, splitting the population into n groups. The asymptotic stability properties are studied.

3. R. B. Kellogg, T. Y. Li and J. A. Yorke, A constructive proof of the Brouwer fixed point theorem and computational results, SIAM J. Numer. Anal. 13 (1976), 473-383.

Abstract: A constructive proof of the Brouwer fixed-point theorem is given, which leads to an algorithm for finding the fixed point. Some properties of the algorithm and some numerical results are also presented.

**1977 **

1. J. L. Kaplan and J. A. Yorke, On the nonlinear differential delay equation dx/dt = -f(x(t), x(t-1)), J. Differential Equations 23 (1977), 293-314.

2. J. L. Kaplan and J. A. Yorke, Competitive exclusion and nonequilibrium coexistence, Amer. Naturalist 111 (1977), 1030-1036.

3. A. Lasota and J. A. Yorke, On the existence of invariant measures for transformations with strictly turbulent trajectories, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astro. Phys.225 (1977), 233-238.

Abstract: A sufficient condition is shown for the existence of nontrivial invariant measures in topological spaces. In particular, it is proved that for any continuous transformation on the real line the existence of a periodic point of period three implies the existence of a continuous invariant measure.

4. J. C. Alexander and J. A. Yorke, Parameterized functions, bifurcation, and vector fields on spheres, Prob. of the Asymptotic Theory of Nonlinear Oscillations Order of the Red Banner, Inst. of Mathematics Kiev 1977, 15-17: Anniversary volume in honor of I. Mitropolsky.

**1978 **

1. T. Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc. 235 (1978), 183-192.

Abstract: A class of piecewise
continuous, piecewise C^{1} transformations on the interval J with
finitely many discontinuities n are shown to have at most n invariant measures

2. J. C. Alexander and J. A. Yorke, Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263-292.

3. S. N. Chow, J. Mallet-Paret and J. A. Yorke, Finding zeroes of maps: Homotopy methods that are constructive with probability one, Math. of Comp. 32 (1978), 887-899.

Abstract: We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is constructive with probability one and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.

4. J. A. Yorke, H. W. Hethcote and A. Nold Dynamics and control of the transmission of gonorrhea, Sexually Transmitted Diseases 5 (1978), 51-56.

5. T. Y. Li and J. A. Yorke, Ergodic maps on [0,1] and nonlinear pseudo-random number generators, Nonlinear Anal. 2 (1978), 473-481.

6. S. N. Chow, J. Mallet-Paret and J. A. Yorke, Global Hopf bifurcation from a multiple eigenvalue, Nonlinear Anal. 2 (1978), 753-763.

7. J. C. Alexander and J. A. Yorke, Calculating bifurcation invariants as elements in the homotopy of the general linear group, J. Pure Appl. Algebra 13 (1978), 1-8.

8. J. C. Alexander and J. A. Yorke, The homotopy continuation method: Numerically implementable topological procedures, Trans. Amer. Math. Soc. 242 (1978), 271-284.

Abstract: The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.

9. T. D. Reynolds, W. P. London and J. A. Yorke, Behavioral rhythms in schizophrenia, J. Nervous and Mental Disease 166 (1978), 489-499.

Abstract: Daily behavioral observations were made for several years on 10 male schizophrenic patients and on three male patients with organic brain disorders. Analysis of these data showed strong cyclic components in the five schizophrenic patients with predominantly hebephrenic symptomatology. Period lengths noted were about 2 days, 5 to 6 day, 30 days, and a longer cycle of 40 to 100 days duration. Antipsychotic medications appear to have a suppressant effect, but tricyclic antidepressants may enhance pre-existing rhythms.

**1979 **

1. J. L. Kaplan, M. Sorg and J. A. Yorke, Solutions of dx/dt = f(x(t), x(t-1)) have limits when f is an order relation, Nonlinear Anal. 3 (1979), 53-58.

2. J. L. Kaplan and J. A. Yorke, Nonassociative real algebras and quadratic differential equations, Nonlinear Anal. 3 (1979), 49-51.

3. J. A. Yorke, N. Nathanson, G. Pianigiani and J. Martin, Seasonality and the requirements for perpetuation and eradication of viruses in populations, Amer. J. Epidemiology 109 (1979), 103-123.

4. G. Pianigiani and J. A. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos, Trans. Amer. Math. Soc. 252 (1979), 351-366.

Abstract: Let A be a subset of R^{n}
be a bounded open set with finitely many connected components A_{j} and
let T be a smooth map on Rn with A a subset of T(A).
Assume that for each A_{j}, A is a subset of T_{k}(Aj) for all
k sufficiently large. We assume that T is expansive, but we do not assume that
T(A) = A. Hence for x in A, T_{i}(x) may escape A as i increases. Let m
be a smooth measure on A (with inf density > 0) and let x in A be chosen at
random (using m). Since T is expansive we may expect T_{i}(x) to
oscillate chaotically on A for a certain time and eventually escape A. For each
measurable set E in A define m_{k}(A) to be the conditional probability
that T_{k}(x) is in E given that x, T_{1}(x), ...,T_{k}
(x) are in A. We show that mk converges to a smooth measure m0 that is independent
of the choice of m which we call a “conditionally invariant measure”.
One-dimensional examples are stressed.

5. J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Comm. Math. Phys. 67 (1979), 93-108. This paper is reprinted in Russian in a book edited by Sinai and Kolmogorov on strange attractors.

Abstract: This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short-term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value that is the first value for which the system possesses a homoclinic orbit.

6. J. A. Yorke and E. D. Yorke Metastable chaos: The transition to sustained chaotic oscillations in a model of Lorenz, J. Stat. Phys. 21 (1979), 263-277.

Abstract: The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbers r somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence upon r is studied both numerically and (very close to the critical r) analytically.

**1980 **

1. J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. 32 (1980), 177-188.

Abstract: We investigate the dynamical properties of continuous maps of a compact metric space into itself. The notion of chaos is defined as the instability of all trajectories in a set together with the existence of a dense orbit. In particular we show that any map on an interval satisfying a generalized period three condition must have a nontrivial (uncountable) minimal set as well as large chaotic subsets. The nontrivial minimal sets are investigated by lifting to sequence spaces while the chaotic sets are investigated using factors, projections of larger spaces onto smaller ones.

**1981 **

1. A. Lasota and J. A. Yorke, The law of exponential decay for expanding mappings , Rend. Sem. Mat. Univ. Padova 64 (1981), 141-157.

2. K. T. Alligood, J. Mallet-Paret and J. A. Yorke, Families of periodic orbits: Local continuability does not imply global continuability, J. Differential Geom. 16 (1981), 483-492.

**1982 **

1. J. Mallet-Paret and J. A. Yorke, Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differential Equations 43 (1982), 419-450.

Abstract: Poincare observed that for a differential equation dx/dt = f(x, a) depending on a parameter a, each periodic orbit generally lies in a connected family of orbits in (x,a)- space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: defined at certain stationary points. We show that generically there are two types of Hopf bifurcation, those we call sources (K = 1) and sinks (K = -1). Generically if the set Q is bounded in (x, a)-space, and if there is an upper bound for periods of the orbits in Q, the Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter snake of orbits. A snake is a maximal path of orbits that contains no orbits whose orbit index is 0.

2. H. W. Hethcote, J. A. Yorke and A. Nold, Gonorrhea modeling: A comparison of control methods, Math. Biosci. 58 (1982), 93-109.

Abstract: A population dynamics model for a heterogeneous population is used to compare the effectiveness of six prevention methods for gonorrhea involving population screening and contact tracing of selected groups. The population is subdivided according to sex, sexual activity, and symptomatic or asymptomatic infection. For this model contact tracing of certain groups is more effective than general population screening.

3. A. Lasota and J. A. Yorke, Exact dynamical systems and the Frobenius-Perron operator. Trans. Amer. Math. Soc. 273 (1982), 375-384.

Abstract: Conditions are investigated that guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps T on [0,1] for which dT/dx(0) > 1. We assume [0,1] can be broken into intervals on which T is continuous and convex and at the left end of these intervals, T = 0 and dT/dx > 0. Such maps have an invariant absolutely continuous density that is exact.

4. T. Y. Li, M. Misiurewicz, G. Pianigiani and J. A. Yorke, Odd chaos, Phys. Lett. 87A (1982), 271-273.

Abstract: The simplest chaotic dynamical processes arise in models that are maps of an interval into itself. Sometimes chaos can be inferred from a few successive data points without knowing the details of the map. Chaos implies knowledge of initial data is insufficient for accurate long term prediction.

5. T. Y. Li, M. Misiurewicz, G. Pianigiani, and J. A. Yorke, No division implies chaos, Trans. Amer. Math. Soc. 273 (1982), 191-199.

Abstract: Let I be a closed interval in R1 and let f be a continuous map on I. Let x0 in I and xi+1 = f (xi) for i 0. We say there is no division for (x1, x2,...,xn) if there is no a in I such that xj < a for all j even and xj < a for all j odd. The main result of this paper proves the simple statement: no division implies chaos. Also given here are some converse theorems, detailed estimates of the existing periods, and examples that show that, under our conditions, one cannot do any better.

6. C. Grebogi, E. Ott and J. A. Yorke, Chaotic attractors in crisis, Phys. Rev. Lett. 48 (1982), 1507-1510. Announcement of #1983-3.

Abstract: The occurrence of sudden qualitative changes of chaotic (or turbulent) dynamics is discussed and illustrated within the context of the one-dimensional quadratic map. For this case, the chaotic region can suddenly widen or disappear, and the cause and properties of these phenomena are investigated.

**1983 **

1. P. Frederickson, J. L. Kaplan, E. D. Yorke and J. A. Yorke, The Lyapunov dimension of strange attractors, J. Differential Equations 49 (1983), 185-207.

Abstract: Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.

2. J. C. Alexander and J. A. Yorke, On the continuability of periodic orbits of parametrized three dimensional differential equations, J. Differential Equations 49 (1983), 171-184.

3. C. Grebogi,

Abstract: The occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated. It is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor. We call such collisions crises. Phenomena associated with crises include sudden changes in the size of chaotic attractors, sudden appearances of chaotic attractors (a possible route to chaos), and sudden destructions of chaotic attractors and their basins. This paper present examples illustrating that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs. In particular the phenomenon of chaotic transients is investigated. The examples discussed illustrate crises in progressively higher dimension and include the one-dimensional quadratic map, the (two-dimensional) Henon map, systems of ordinary differential equations in three dimensions and a three-dimensional map. In the case of our study of the three-dimensional map a new route to chaos is proposed that is possible only in invertible maps or flows of dimension at least three or four, respectively. Based on the examples presented the following conjecture is proposed: almost all sudden changes in the size of chaotic attractors and almost all sudden destructions or creations of chaotic attractors and their basins are due to crises.

4. J. D. Farmer,

Abstract: Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the dimension of the natural measure, and all of the metric dimensions take on a common value, which we call the fractal dimension. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.

5. C. Grebogi, E. Ott and J. A. Yorke, Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation, Phys. Rev. Lett. 50 (1983), 935-938, E 51 (1983), 942.

Abstract: A new type of bifurcation to chaos is pointed out and discussed. In this bifurcation two unstable fixed points or periodic orbits are created simultaneously with a strange attractor that has a fractal basin boundary. Chaotic transients associated with the coalescence of the unstable-unstable pair are shown to be extraordinarily long-lived.

6. C. Grebogi, E. Ott and J. A. Yorke, Are three frequency quasiperiodic orbits to be expected in typical nonlinear dynamical systems?, Phys. Rev. Lett. 51 (1983), 339-342. Announcement of #1985-4.

Abstract: The current state of theoretical understanding related to the question posed in the title is incomplete. This paper presents results of numerical experiments that are consistent with a positive answer. These results also bear on the problem of characterizing possible routes to chaos in nonlinear dynamical systems.

7. J. A. Yorke and K. T. Alligood Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull. Amer. Math. Soc. 9 (1983), 319-322. Announcement of #1985-7.

8. C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Final state sensitivity: An obstruction to predictability, Phys. Letters 99A (1983), 415-418.

Abstract: It is shown that nonlinear systems with multiple attractors commonly require very accurate initial conditions for the reliable prediction of final states. A scaling exponent for the final-state-uncertain phase space volume dependence on uncertainty in initial conditions is defined and related to the fractal dimension of basin boundaries.

**1984 **

1. J. L. Kaplan, J. Mallet-Paret and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory and Dyn. Sys. 4 (1984), 261-281.

Abstract: The fractal dimension of an attracting torus Tk in R X Tk is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1.

2. B. Curtis Eaves and J. A. Yorke, Equivalence of surface density and average directional density, Math. Operations Res. 9 (1984), 363-375.

Abstract: The average directional density criteria for evaluating tilings is shown to be equivalent to surface density and valid for random broken paths just as for straight paths.

3. K. T. Alligood and J. A. Yorke, Families of periodic orbits: Virtual periods and global continuability, J. Differential Equations 55 (1984), 59-71.

Abstract: For a differential equation depending on a parameter, there have been numerous investigations of the continuation of periodic orbits as the parameter is varied. Mallet-Paret and Yorke investigated in generic situations how connected components of orbits must terminate. Here we extend the theory to the general case, dropping genericity assumptions.

4. J. C. Alexander and J. A. Yorke, Fat baker's transformations, Ergodic Theory and Dyn. Sys. 4 (1984), 1-23.

Abstract: We investigate a variant of the baker transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated. Erdos' results get a geometric/ dynamical systems realization. P.S.: In 1995 Solomniak showed that for almost every expansion constant the measure is 2 dimensional, solving a problem begun by Erdos.

5. B. R. Hunt and J. A. Yorke, When all solutions of dx/dt = Σi qi(t)x(t-Ti(t)) oscillate, J. Differential Equations 53 (1984), 139-145.

Abstract: In this paper the long-term behavior of solutions to the equation in the title are examined, where qi(t) and Ti(t) are positive. In particular, it is shown that if liminf sumi = ln Ti(t)qi(t) > 1/ e, all solutions oscillate about 0 infinitely often.

6. A. Lasota, T. Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751-764.

Abstract: We say the operator P on L1 is a Markov operator if (i) Pf ≥ 0 for f ≥ 0 and (ii) |Pf| = |f| if f ≥ 0. It is shown that any Markov operator P has certain spectral decomposition if, for any f in L1 with f = 0 and the norm of f = 1, Pnf converges to f when n goes to infinity, where F is a strongly compact subset of L1. It follows from this decomposition that Pnf is asymptotically periodic for any f in L1.

7. C. Grebogi, E. Ott, S. Pelikan and J. A. Yorke, Strange attractors that are not chaotic, Physica 13D (1984), 261-268.

Abstract: It is shown that in certain types of dynamical systems it is possible to have attractors that are strange but not chaotic. Here we use the work strange to refer to the geometry or shape of the attracting set, while the word chaotic refers to the dynamics of orbits on the attractor (in particular, the exponential divergence of nearby trajectories). We first give examples for which it can be demonstrated that there is a strange nonchaotic attractor. These examples apply to a class of maps that model nonlinear oscillators (continuous time) that are externally driven at two incommensurate frequencies. It is then shown that such attractors are persistent under perturbations that preserve the original system type (i.e., there are two incommensurate external driving frequencies). This suggests that, for systems of the type that we have considered, nonchaotic strange attractors may be expected to occur for a finite interval of parameter values. On the other hand, when small perturbations that do not preserve the system type are numerically introduced, the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit. Thus we conjecture that, in general, continuous time systems that are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space.

8. E. Ott, W. D. Withers and J. A. Yorke, Is the dimension of chaotic attractors invariant under coordinate changes?, J. Stat. Phys. 36 (1984), 687-697.

Abstract: Several different dimension-like quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except as a finite number of points. It is found that some are and some are not. It is suggested that the word dimension be reversed only for those quantities that have this invariance property.

**1985 **

1. T. Y. Li, J. Mallet-Paret and J. A. Yorke, Regularity results for real analytic homotopies, Numerische Mathematik 46 (1985), 43-50.

Abstract: In this paper, we study two main features of the homotopy curves that we follow when we use the homotopy method for solving the zeros of analytic maps. First, we prove that near the solution the curve behaves nicely. Secondly, we prove that the set of starting points that give smooth homotopy curves is open and dense. The second property is of particular importance in computer implementation.

2. E. Ott, E. D. Yorke and J. A. Yorke, A scaling law: How an attractor's volume depends on noise level, Physica 16D (1985), 62-78.

Abstract: We investigate the meaning of the dimension of strange attractor for systems with noise. More specifically, we investigate the effect of adding noise of magnitude g to a deterministic system with D degrees of freedom. If the attractor has dimension d and d < D, then its volume is zero. The addition of noise may be an important physical probe for experimental situations, useful for determining how much of the observed phenomena in a system is due to noise already present. When the noise is added the attractor Ag has positive volume. We conjecture that the generalized volume of Ag is proportional to gD-d for g near 0 and show this relationship is valid in several cases. For chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly.

3. J. A. Yorke, C. Grebogi, E. Ott and L. Tedeschini-Lalli Scaling behavior of windows in dissipative dynamical systems, Phys. Rev. Lett. 54 (1985), 1095-1098.

Abstract: Global scaling behavior for period-n windows of chaotic dynamical systems is demonstrated. This behavior should be discernible in experiments.

4. C. Grebogi, E. Ott and J. A. Yorke, Attractors on an N-torus: Quasiperiodicity versus chaos, Physica 15D (1985), 354-373. Announcement in #1983-6.

Abstract: The occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance. However, current understanding concerning the question of how prevalent such motions should be is incomplete With this in mind, the types of attractors that can exist for flows on the N - torus are studied numerically for N = 3 and 4. Specifically, nonlinear perturbations are applied to maps representing N - frequency quasiperiodic attractors. These perturbations can cause the original N - frequency quasiperiodic attractors to bifurcate to other types of attractors. Our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows: N - frequency quasiperiodic attractors are the most common, followed by (N - 1)- frequency quasiperiodic attractors,..., followed by period attractors. However, as the nonlinearity is further increased, N-frequency quasiperiodicity becomes less common, ceasing to occur when the map becomes noninvertible. Chaotic attractors are very rare for N = 3 for small to moderate nonlinearity, but are somewhat more common for N = 4. Examination of the types of chaotic attractors that occur for N = 3 reveals a rich variety of structure and dynamics. In particular, we see that there are chaotic attractors that apparently fill the entire N - torus (i.e., limit sets of orbits on these attractors are the entire torus); furthermore, these are the most common types of chaotic attractors at moderate nonlinearities.

5. S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, Fractal basin boundaries, in Physica 17D (1985), 125-153.

Abstract: Basin boundaries for dynamical systems can be either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structure are given and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.

6. S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, Structure and crises of fractal basin boundaries, Phys. Lett. 107A (1985), 51-54.

Abstract: We discuss the structure of fractal basin boundaries in typical nonanalytic maps of the plane and describe a new type of crisis phenomenon.

7. J. A. Yorke and K. T. Alligood, Period doubling cascades of attractors: A prerequisite for horseshoes, Comm. Math. Phys. 101 (1985), 305-321. Announcement in #1983-7. See also #1987-8.

Abstract: This paper shows that if a horseshoe is created in a natural manner as a parameter is varied, then the process of creation involves the appearance of attracting periodic orbits of all periods. Furthermore, each of these orbits will period double repeatedly, with those periods going to infinity.

8. C. Grebogi, E. Ott and J. A. Yorke, Super persistent chaotic transients, Ergodic Theory and Dyn. Sys. 5 (1985), 341-372.

Abstract: The unstable-unstable pair bifurcation is a bifurcation in which two unstable fixed points of periodic orbits of the same period coalesce and disappear as a system parameter is raised. For parameter values just above that at which unstable orbits are destroyed there can be chaotic transients. Then, as the bifurcation is approached from above, the average length of a chaotic transient diverges, and, below the bifurcation point, the chaotic transient may be regarded as having been converted into a chaotic attractor. It is argued that unstable-unstable pair bifurcations should be expected to occur commonly in dynamical systems. This bifurcation is an example of the crisis route to chaos. The most striking fact about unstable-unstable pair bifurcation crises is that long chaotic transients persist even for parameter values relatively far from the bifurcation point. These long-lived chaotic transients may prevent the time asymptotic state from being reached during experiments. An expression giving a lower bound for the average lifetime of a chaotic transient is derived and shown to agree well with numerical experiments. In particular, this bound on the average lifetime, T, satisfies

T = k1 exp [k2(a-a0)-1/ 2]

for a near a0, where k1 and k2 are constants and a0 is the value of the parameter a at which the crisis occurs. Thus, as a approaches a0 from above, T increases more rapidly than any power of (a-a0)-1. Finally, we discuss the effect of adding bounded noise (small random perturbations) on these phenomena and argue that the chaotic transient should be lengthened by noise.

** **

** **

**1986 **

1. C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, The exterior dimension of fat fractals, Phys. Lett. 110A (1985), 1-4; E 113A (1986), 495. Also, Comment on "Sensitive dependence on parameters in nonlinear dynamics" and on "Fat fractals on the energy surface" (with C. Grebogi and E. Ott), Phys. Rev. Lett 56 (1986), 266.

Abstract: Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity that we call the exterior dimension. In addition, it is shown that the exterior dimension is related to the uncertainty exponent previously used in studies of fractal basin boundaries, and it is show how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.

2. C. Grebogi, E. Ott and J. A. Yorke, Metamorphoses of basin boundaries in nonlinear dynamical systems, Phys. Rev. Lett. 56 (1986), 1011-1014.

Abstract: A basin boundary can undergo sudden changes in its character as a system parameter passes through certain critical values. In particular, basin boundaries can suddenly jump in position and can change from being smooth to being fractal. We describe these changes (metamorphoses) and find that they involve certain special unstable orbits on the basin boundary that are accessible from inside one of the basins. The forced damped pendulum (Josephson junction) is used to illustrate these phenomena.

3. A. Lasota and J. A. Yorke, Statistical Periodicity of Deterministic Systems, Casopis Pro Pestovani Matematiky 111 (1986), 1-13.

4. K. T. Alligood and J. A. Yorke, Hopf bifurcation: The appearance of virtual periods in cases of resonance, J. Differential Equations 64 (1986), 375-394.

5. L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys. 106 (1986), 635-657.

Abstract: This work concerns the
nature of chaotic dynamical processes. Sheldon Newhouse wrote on dynamical
processes (depending on a parameter m) x_{n+1 }= T(x_{n}; m),
where x is in the plane, such as might arise when studying Poincare return maps
for autonomous differential equations in R^{3}. He proved that if the
system is chaotic there will very often be existing parameter values for which
there are infinitely many periodic attractors coexisting in a bounded region of
the plane, and that such parameter values m would be dense in some interval.
The fact that infinitely many coexisting sinks can occur brings into question
the very nature of the foundations of chaotic dynamical processes. We prove,
for an apparently typical situation, that the Newhouse construction yields only
a set of parameter values m of measure zero.

6. C. Grebogi, E. Ott and J. A. Yorke, Critical exponent of chaotic transients in nonlinear dynamical systems, Phys. Rev. Lett. 57 (1986), 1284-1287.

Abstract: The average lifetime of
a chaotic transient versus a system parameter is studied for the case wherein a
chaotic attractor is converted into a chaotic transient upon collision with its
basin boundary (a crisis). Typically the average lifetime T depends upon the
system parameter p via T is proportional [p-p_{0}]-g, where p_{0}
denotes the value of p at the crisis. A theory determining g for
two-dimensional maps is developed and compared with numerical experiments. The
theory also applies to critical behavior at interior crises.

7. J. L. Hudson, O. E. Rossler and J. A. Yorke, Cloud attractors and time-inverted Julia boundaries, Z. Naturforsch 41A (1986), 979-980.

**1987 **

1. C. Grebogi, E. Kostelich, E. Ott and J. A. Yorke, Multi-dimensioned intertwined basin boundaries and the kicked double rotor, Phys. Letters 118A (1986), 448-454; E 120A (1987), 497.

Abstract: Using two examples, one a four dimensional kicked double rotor and the other a simple noninvertible one dimensional map, we show that basin boundary dimensions can be different in different regions of phase space. For example, they can be fractal or not fractal depending on the region. In addition, we show that these regions of different dimension can be intertwined on arbitrarily fine scale. We conjecture, based on these examples, that a basin boundary typically can have at most a finite number of possible dimension values.

2. E. Kostelich and J. A. Yorke, Lorenz cross sections of the chaotic attractor of the double rotor, Physica 24D (1987), 263-278.

Abstract: A Lorenz cross section of an attractor with k > 0 positive Lyapunov exponents arising from a map of n variables is the transverse intersection of the attractor with an (n - k)-dimensional plane. We describe a numerical procedure to compute Lorenz cross sections of chaotic attractors with k > 1 positive Lyapunov exponents and apply the technique to the attractor produced by the double rotor map, two of whose numerically computed Lyapunov exponents are positive and whose Lyapunov dimension is approximately 3.64. Error estimates indicate that the cross sections can be computed to high accuracy. The Lorenz cross sections suggest that the attractor for the double rotor map locally is not the cross product of two intervals and two Cantor sets. The numerically computed pointwise dimension of the Lorenz cross sections is approximately 1.64 and is independent of where the cross section plane intersects the attractor. This numerical evidence supports a conjecture that the pointwise and Lyapunov dimensions of typical attractors are equal.

3. J. A. Yorke, E. D. Yorke, and J. Mallet-Paret, Lorenz-like chaos in a partial differential equation for a heated fluid loop, Physica 24D (1987), 279-291.

Abstract: A set of partial differential equations are developed describing fluid flow and temperature variation in a thermosyphon with particularly simple external heating. Several exact mathematical results indicate that a Bessel-Fourier expansion should converge rapidly to a solution. Numerical solutions for the time-dependent coefficients of that expansion exhibit a transition to chaos like that shown by the Lorenz equations over a wide range of fluid material parameters.

4. T. Y. Li, T. Sauer and J. A. Yorke, Numerical solution of a class of deficient polynomial systems, SIAM J. Numer. Anal. 24 (1987), 435-451.

Abstract: Most systems of polynomials that arise in applications have fewer than the expected number of solutions. The amount of computation required to find all solutions of such a deficient system using current homotopy continuation methods is proportional to the expected number of solutions and, roughly, to the size of the system. Much time is wasted following paths that do not lead to solutions. We suggest methods for solving some deficient polynomial systems for which the amount of computational effort is instead proportional to the number of solutions.

5. C. Grebogi, E. Ott and J. A. Yorke, Basin boundary metamorphoses: Changes in accessible boundary orbits, Physica 24D (1987), 243-262, and Nucl. Phys. B. (Suppl.) 2 (1987), 281-300.

Abstract: Basin boundaries sometimes undergo sudden metamorphoses. These metamorphoses can lead to the conversion of a smooth basin boundary to one that is fractal, or else can cause a fractal basin boundary to suddenly jump in size and change its character (although remaining fractal). For an invertible map in the plane, there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal. Nonetheless, we have found that typically only one of them can be reached or accessed directly from a given basin. The other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary. The boundary metamorphoses that we investigate are characterized by a sudden replacement of the basin boundary's accessible orbit.

6. C. Grebogi, E. Ott and J. A. Yorke, Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics, Science 238 (1987), 632-638.

Abstract: Recently research has shown that many simple nonlinear deterministic systems can behave in an apparently unpredictable and chaotic manner. This realization has broad implications for many fields of science. Basic developments in the field of chaotic dynamics of dissipative systems are reviewed in this article. Topics covered include strange attractors, how chaos comes about with variation of a system parameter, universality, fractal basin boundaries and their effect on predictability, and applications to physical systems.

7. C. Grebogi, E. Kostelich, E. Ott and J. A. Yorke, Multi-dimensioned intertwined basin boundaries: Basin structure of the kicked double rotor, Physica 25D (1987), 347-360.

Abstract: Using numerical
computations on a map that describes the time evolution of a particular
mechanical system in a four-dimensional phase space (The kicked double rotor),
we have found that the boundaries separating basins of attraction can have
different properties in different regions and that these different regions can
be intertwined on arbitrarily fine scale. In particular, for the double rotor
map, if one chooses a restricted region of the phase space and examines the
basin boundary in that region, then either one observes that the boundary is a
smooth three-dimensional surface or one observes that the boundary is fractal
with dimension d ~ 3.9, and which of these two possibilities applies depends on
the particular phase space region chosen for examination. Furthermore, for any
region (no matter how small) for which d ~ 3.9, one can choose subregions
within it for which d = 3. (Hence d ~ 3.9 region and d = 3 region are
intertwined on arbitrarily fine scale.) Other examples will also be presented
and analyzed to show how this situation can arise. These include
one-dimensional map cases, a map of the plane and the Lorenz equations. In one
of our one-dimensional map cases the boundary will be fractal everywhere, but
the dimension can take on either of two different values both of which lie
between 0 and 1. These examples lead us to conjecture that basin boundaries
typically can have at most a finite number of possible dimension values. More
specifically, let these values be denoted d_{1}, d_{2},...,d_{N}.
Choose a volume region of phase space whose interior contains some part of the
basin boundary and evaluate the dimension of the boundary in that region. Then
our conjecture is that for all typical volume choices, the evaluated dimension
within the chosen volume will be one of the values d_{1}, d_{2},...,d_{N}.
For example, in our double rotor map it appears that N = 2, and d_{1} =
3.0 and d_{2} = 3.9.

8. K. T. Alligood, E. D. Yorke and J. A. Yorke, Why period-doubling cascades occur: Periodic orbit creation followed by stability shedding, Physica 28D (1987), 197-205.

Abstract: Period-doubling cascades of attractors are often observed in low-dimensional systems prior to the onset of chaotic behavior. We investigate conditions that guarantee that some kinds of cascades must exist.

9. C. Grebogi, E. Ott and J. A. Yorke, Unstable periodic orbits and the dimension of chaotic attractors, Phys. Rev. A, 36 (1987), 3522-3524.

Abstract: A formulation giving the q dimension Dq of a chaotic attractor in terms of the eigenvalues of unstable periodic orbits is presented and discussed.

10. F. Varosi, C. Grebogi and J. A. Yorke, Simplicial approximation of Poincare maps of differential equations, Phys. Letters A124 (1987), 59-64.

Abstract: A method is proposed to transform a nonlinear differential system into a map without having to integrate the whole orbit as in the usual Poincare return map technique. It consists of constructing a piecewise linear map by coarse-graining the phase surface of section into simplices and using the Poincare return map values at the vertices to define a linear map on each simplex. The numerical results show that the simplicial map is a good approximation to the Poincare map and it leads to a factor of 20 to 40 savings in computer time as compared with the integration of the differential equation. Computation of the generalized information dimensions of a chaotic orbit for the simplicial map gives values in close agreement with those found for the Poincare map.

11. S. M. Hammel, J. A. Yorke and C. Grebogi, Do numerical orbits of chaotic dynamical processes represent true orbits?, J. of Complexity 3 (1987), 136-145.

Abstract: Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. In an iterated process, if errors double each iterate and numerical calculations have 50-bit (or 15-digit) accuracy, a true orbit through a point can be expected to have no correlation with a numerical orbit after 50 iterates. On the other hand, numerical studies often involve hundreds of thousands of iterates. One may therefore question the validity of such studies. A relevant result in this regard is that of Anosov and Bowen who showed that systems that are uniformly hyperbolic will have the shadowing property: a numerical (or noisy) orbit will stay close to (shadow) a true orbit for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report rigorous results for nonhyperbolic systems: numerical orbits typically can be shadowed by true orbits for long time periods.

12. C. Grebogi, E. Ott, J. A. Yorke and H. E. Nusse, Fractal basin boundaries with unique dimension, Ann. N.Y. Acad. Sci 497, (1987), 117-126.

13. T. Y. Li, T. Sauer and J. A. Yorke, The random product homotopy and deficient polynomial systems, Numerische Mathematik 51 (1987), 481-500.

Abstract: Most systems of n polynomial equations in n unknowns arising in applications are deficient, in the sense that they have fewer solutions than that predicted by the total degree of the system. We introduce the random product homotopy, an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems. In many cases, the amount of computation required to find all solutions can be made roughly proportional to the number of solutions.

14. C. Grebogi, E. Ott, F. Romeiras and J. A. Yorke, Critical exponents for crisis induced intermittency, Phys. Rev. A 36 (1987), 5365-5380.

Abstract: We consider three
types of changes that attractors can undergo as a system parameter is varied.
The first type leads to the sudden destruction of a chaotic attractor. The
second type leads to the sudden widening of a chaotic attractor. In the third
type of change, which applies for many systems with symmetries, two (or more)
chaotic attractors merge to form a single chaotic attractor and the merged
attractor can-be larger in phase-space extent than the union of the attractors
before the change. All three of these types of changes are termed crises and
are accompanied by a characteristic temporal behavior of orbits after the
crisis. For the case where the chaotic attractor is destroyed, this
characteristic behavior is the existence of chaotic transients. For the case
where the chaotic attractor suddenly widens, the characteristic behavior is an
intermittent bursting out of the phase-space region within that the attractor
was confined before the crisis. For the case where the attractors suddenly
merge, the characteristic behavior is an intermittent switching between
behaviors characteristic of the attractors before merging. In all cases a time
scale T can be defined that quantifies the observed post-crisis behavior: for
attractor destruction, T is the average chaotic transient lifetime; for
intermittent bursting, it is the mean time between bursts; for intermittent
switching it is the mean time between switches. The purpose of this paper is to
examine the dependence of T on a system parameter (call it p) as this parameter
passes through its crisis value p = p_{c}. Our main result is that for
an important class of systems the dependence of T on p is T is proportional to
p-p_{c} raised to a power g for p close to p_{c} , and we
develop a quantitative theory for the determination of the critical exponent g.
Illustrative numerical examples are given. In addition, applications to
experimental situation, as well as generalizations to higher-dimensional cases,
are discussed. Since the case of attractor destruction followed by chaotic
transients has previously been illustrated with examples [C. Grebogi, E. Ott
and J. A. Yorke, Phys. Rev. Lett. 57, 1284 (1986)], the numerical examples
reported in this paper will be for crisis-induced intermittency (i.e.,
intermittent bursting and switching).

**1988 **

1. C. Grebogi, E. Ott and J. A. Yorke, Unstable periodic orbits and the dimensions of multifractal chaotic attractors, Phys. Rev. A 37 (1988), 1711-1724.

Abstract: The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily fine-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and chaotic repellers are considered.

2. E. M. Coven, I. Kan and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc. 308 (1988), 227-241.

Abstract: We study the family of tent maps - continuous, unimodal, piecewise linear maps of the interval with slopes absolute value s, sqrt (2) ≤ s ≤2. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters s, although they fail to have the shadowing property for an uncountable, dense set of parameters. We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.

3. H. E. Nusse and J. A. Yorke, Is every approximate trajectory of some process near an exact trajectory of a nearby process?, Comm. Math. Phys. 114 (1988), 363-379.

Abstract: This paper deals with the problem: Can a noisy orbit be tracked by a real orbit? In particular, we will study the one-parameter family of tent maps and the one-parameter family of quadratic maps. We write gm for either fm or Fm with fm(x) = mx for x ≤ 1/2 and fm (x) = m(1-x) for x ≥ 1/2, and Fm(x) = mx (1-x). For a given m we will say: gm permits “increased parameter shadowing” if for each delta-x > 0 there exists some delta-m > 0 and some delta-f > 0 such that every delta-f -pseudo gm-orbit starting in some invariant interval can be deltax -shadowed by a real ga -orbit with a = m + delta m. We show that gm typically permits increased parameter shadowing.

4. H. E. Nusse and J. A. Yorke, Period halving for xn+1 = MF(xn) where F has negative Schwarzian derivative, Phys. Letters A 127 (1988), 328-334.

Abstract: We present an example of a one-parameter family of maps F (x; m) = mF(x) where the map F if unimodal and has a negative Schwarzian derivative. We will show for our example that (1) some regular period-halving bifurcations do occur and (2) the topological entropy can decrease as the parameter m is increased.

5. E. Kostelich and J. A. Yorke, Noise reduction in Dynamical Systems, Phys. Rev. A. 38 (1988), 1649-1652.

Abstract: A method is described for reducing noise levels in certain experimental time series. An attractor is reconstructed from the data using the time-delay embedding method. The method produces a new, slightly altered time series that is more consistent with the dynamics on the corresponding phase-space attractor. Numerical experiments with the two-dimensional Ikeda laser map and power spectra from weakly turbulent Couette-Taylor flow suggest that the method can reduce noise levels up to a factor of 10.

6. S. M. Hammel, J. A. Yorke and C. Grebogi, Numerical orbits of chaotic processes represent true orbits, Bull. Amer. Math. Soc. 19 (1988), 465-469.

7. P. M. Battelino, C. Grebogi, E. Ott, J. A. Yorke and E. D. Yorke Multiple coexisting attractors, basin boundaries and basic sets, Physica 32 D (1988), 296-305.

Abstract: Orbits initialized exactly on a basin boundary remain on that boundary and tend to a subset on the boundary. The largest ergodic such sets are called basic sets. In this paper we develop a numerical technique that restricts orbits to the boundary. We call these numerically obtained orbits straddle orbits. By following straddle orbits we can obtain all the basic sets on a basin boundary. Furthermore, we show that knowledge of the basic sets provides essential information on the structure of the boundaries. The straddle orbit method is illustrated by two systems as examples. The first system is a damped driven pendulum that has two basins of attraction separated by a fractal basin boundary. In this case the basic set is chaotic and appears to resemble the product of two Cantor sets. The second system is a high-dimensional system (five phase space dimensions), namely, two coupled drive Van der Pol oscillators. Two parameter sets are examined for this system. In one of these cases the basin boundaries are not fractal, but there are several attractors and the basins are tangled in a complicated way. In this case all the basin sets are found to be unstable periodic orbits. It is then shown that using the numerically obtained knowledge of the basic sets, one can untangle the topology of the basin boundaries in the five-dimensional phase space. In the case of the other parameter set, we find that the basin boundary is fractal and contains at least two basic sets one of which is chaotic and the other quasiperiodic.

8. C. Grebogi,

Abstract: Due to roundoff, digital computer simulations of orbits on chaotic attractors will always eventually become periodic. The expected period, probability distribution of periods, and expected number of periodic orbits are investigated for the case of fractal chaotic attractors. The expected period scales with roundoff epsilon as epsilon-d/2, where d is the correlation dimension of the chaotic attractor.

9. T. Y. Li, T. Sauer, J. A. Yorke, Numerically determining solutions of systems of polynomial equations, Bull. Amer. Math. Soc. 18 (1988), 173-177.

**1989 **

1. I. Kramer, E. D. Yorke and J. A. Yorke, The AIDS epidemic's influence on the gay contact rate from analysis of gonorrhea incidence, Math. Comput. Modelling 12 (1989), 129-137.

Abstract: To model the AIDS
epidemic in the homosexual population it is necessary to determine the
time-dependent decrease in the unprotected contact rate caused by awareness of
AIDS. The San Francisco

2. E. Ott, C. Grebogi and J. A. Yorke, Theory of first order phase transitions for chaotic attractors of nonlinear dynamical systems, Phys. Letters A 135 (1989), 343-348.

Abstract: A theory is presented for first order phase transitions of multifractal chaotic attractors of nonhyperbolic two-dimensional maps. (These phase transitions manifest themselves as a discontinuity in the derivative with respect to q (analogous to temperature) of the fractal dimension q-spectrum, Dq (analogous to free energy).) A complete picture of the behavior associated with the phase transition is obtained.

3. E. Ott, T. Sauer and J. A. Yorke, Lyapunov partition functions for the dimensions of chaotic sets, Phys. Rev. Lett. A 39 (1989), 4212-4222.

Abstract: Multifractal dimension spectra for the stable and unstable manifolds of invariant chaotic sets are studied for the case of invertible two-dimensional maps. A dynamical partition-function formalism giving these dimensions in terms of local Lyapunov numbers is obtained. The relationship of the Lyapunov partition functions for stable and unstable manifolds to previous work is discussed. Numerical experiments demonstrate that dimension algorithms based on the Lyapunov partition function are often very efficient. Examples supporting the validity of the approach for hyperbolic chaotic sets and for nonhyperbolic sets below the phase transition(q<qT) are presented.

4. T. Y. Li, T. Sauer and J. A. Yorke, The cheater's homotopy: An efficient procedure for solving systems of polynomial equations, SIAM J. Numer. Anal. 26 (1989), 1241-1251. Also announcement: Bull. Amer. Math. Soc. 18 (1988), 173-177: Numerically determining solutions of systems of polynomial equations.

Abstract: A procedures is introduced for solving systems of polynomial equations that need to be solved repetitively with varying coefficients. The procedure is based on the cheater homotopy, a continuation method that follows paths to all solutions. All solutions are found with an amount of computational work roughly proportional to the actual number of solutions. Previous general methods normally require an amount of computation roughly proportional to the total degree.

5. H. E. Nusse and J. A. Yorke, A procedure for finding numerical trajectories on chaotic saddles, Physica D 36 (1989), 137-156.

Abstract: Examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors. If almost every trajectory eventually leaves some regions, but the region contains a chaotic set, then typical trajectories will behave chaotically for a while and then will leave the region, and so we will observe chaotic transients. The main objective that will be addressed is the Dynamic Restraint Problem: Given a region that contains a chaotic set but does not contain an attractor, find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time. Systems with horseshoes have such regions as do systems with fractal basin boundaries, as does the Henon map for suitable chosen parameters. We present a numerical technique for finding trajectories that will stay in such chaotic sets for arbitrarily long periods of time and it leads to a saddle straddle trajectory. The method is called the PIM triple procedure since it is based on so-called PIM triple. A PIM (Proper Interior Maximum) triple is three point (a, c, b) in a straight line segment such that the interior point c (i.e. c is between a and b) has the maximum escape time, that is, its escape time from the region is greater than the escape time of both a and b. Proper means the segment from a to b is smaller than a previously obtained segment. We show rigorously that the PIM triple procedure works in ideal situations. We find it works well even in less than ideal cases. This procedure can also be used for the computation of Lyapunov exponents.

Furthermore, the accessible PIM triple procedure (a refined PIM triple procedure for finding accessible trajectories on the chaotic saddle) will also be discussed.

6. P. M. Battelino, C. Grebogi, E. Ott and J. A. Yorke, Chaotic attractors on a 3-torus and torus break-up, Physica D 39 (1989), 299-314.

Abstract: Two coupled driven Van der Pol oscillators can have three-frequency quasiperiodic attractors, which lie on a 3-torus. The evidence presented in this paper indicates that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent. Furthermore, no chaotic orbits lying on a torus were observed, suggesting that, in most cases, at least in the case of this system, orbits do not become chaotic before their tori are destroyed. To expedite the calculations, a method was developed, which can be used to determine if an orbit is on a torus, without actually displaying that orbit. The method, also described in this paper, was designed specifically for our system. The basic idea, however, could be used for studying attractors of other systems. Very few modifications of the method, if any, would be necessary when studying systems with the number of degrees of freedom equal to that of our Van der Pol system.

7. B-S. Park, C. Grebogi, E. Ott and J. A. Yorke, Scaling of fractal basin boundaries near intermittency transitions to chaos, Phys. Rev. A 40 (1989), 1576-1581.

Abstract: It is the purpose of
this paper to point out that the creation of fractal basin boundaries is a
characteristic feature accompanying the intermittency transition to chaos.
(Here intermittency transition is used in the sense of Pomeau and Manneville
[Commun. Math. Phys. 74, 189 (1980)]; viz., a chaotic attractor is created as a
periodic orbit becomes unstable.) In particular, we are here concerned with
type-I and type-_{0}-k[p-pI], where d_{0}
is the dimension at the intermittency transition parameter value p = pI and k
is a scaling constant. Furthermore, for type-I intermittency d_{0} <
D, while for type-

8. W. L. Ditto, S. Rauseo, R. Cawley, C. Grebogi, G. H. Hsu, E. Kostelich, E. Ott, H. T. Savage, R. Segnan, M. Spano and J. A. Yorke, Experimental observation of crisis-induced intermittency and its critical exponent, Phys. Rev. Lett. 63 (1989), 923-926.

Abstract: Critical behavior
associated with intermittent temporal bursting accompanying the sudden widening
of a chaotic attractor was observed and investigated experimentally in a
gravitationally buckled, parametrically driven, magnetoelastic ribbon. As the
driving frequency, f, was decreased through the critical value, f_{c},
we observed that the mean time between bursts scaled as the absolute value of f_{c}
- f to a power of -g.

9. E. J. Kostelich and J. A. Yorke, Using dynamic embedding methods to analyze experimental data, Contemp. Math. 99 (1989), 307-312.

Abstract: The time-delay embedding method provides a powerful tool for the analysis of experimental data. We show how recent improvements allow experimentalists to use many of the same techniques that have been essential to the analysis of nonlinear systems of ordinary differential equations and difference equations.

** **

**1990 **

1. I. Kramer, E. D. Yorke and J. A. Yorke, Modelling non-monogamous heterosexual transmission of AIDS, Math. Comput. Modelling 13 (1990) 99-107.

2. E. Kostelich and J. A. Yorke, Noise reduction: Finding the simplest dynamical system consistent with the data, Physica D 41 (1990), 183-196.

Abstract: A novel method is described for noise reduction in chaotic experimental data whose dynamics are low dimensional. In addition, we show how the approach allows experimentalists to use many of the same techniques that have been essential for the analysis of nonlinear systems of ordinary differential equations and difference equations.

3.

Abstract: One-parameter families of ƒλ
of diffeomorphisms of the Euclidean plane are known to have a complicated
bifurcation pattern as λ varies near certain values, namely where
homoclinic tangencies are created. We
argue that the bifurcation pattern is much more irregular than previously
reported. Our results contrast with the
monotonicity result for the well-understood one-dimensional family g_{λ}(x) = λx(1
– x), where it is known that periodic orbits are created and never annihilated
as λ increases. We show that this
monotonicity in the creation of periodic orbits never occurs for any
one-parameter family of C^{3} area contracting diffeomorphisms of the
Euclidean plane, excluding certain technical degenerate cases where our
analysis breaks down. It has been shown
that in each neighborhood of a parameter value at which a homoclinic tangency
occurs, there are either infinitely many parameter values at which periodic
orbits are created or infinitely many at which periodic orbits are
annihilated. We show that there are both
infinitely many values at which periodic orbits are created and infinitely many
at which periodic orbits are annihilated.
We call this phenomenon antimonotonicity

4. C. Grebogi, S. M. Hammel, J. A. Yorke and T. Sauer, Shadowing of physical trajectories in chaotic dynamics: Containment and refinement, Phys. Rev. Lett. 65 (1990), 1527-1530.

Abstract: For a chaotic system, a noisy trajectory diverges rapidly from the true trajectory with the same initial condition. To understand in what sense the noisy trajectory reflects the true dynamics of the actual system, we developed a rigorous procedure to show that some true trajectories remain close to the noisy one for long times. The procedure involves a combination of containment, which establishes the existence of an uncountable number of true trajectories close to the noisy one, and refinement, which produces a less noisy trajectory. Our procedure is applied to noisy chaotic trajectories of the standard map and the driven pendulum.

5. T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Lett. 65 (1990), 3215-3218.

Abstract: A method is developed that uses the exponential sensitivity of a chaotic system to tiny perturbations to direct the system to a desired accessible state in a short time. This is done by applying a small, judiciously chosen, perturbation to an available system parameter. An expression for the time required to reach an accessible state by applying such a perturbation is derived and confirmed by numerical experiment. The method introduced is shown to be effective even in the presence of small-amplitude noise or small modeling errors.

6. E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (1990), 1196-1199.

Abstract: It is shown that one can convert a chaotic attractor to any one of a large number of possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter. The method utilizes delay coordinate embedding, and so is applicable to experimental situations in which a priori analytical knowledge of the system dynamics is not available. Important issues include the length of the chaotic transient preceding the periodic motion, and the effect of noise. These are illustrated with a numerical example.

7. M. Ding, C. Grebogi, E. Ott and J. A. Yorke, Transition to chaotic scattering, Phys. Rev. A, 42 (1990), 7025-7040.

Abstract: This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e., not chaotic). Our results show that the transition from regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of fully developed chaotic scattering can be reached in our system through a process analogous to the formation of a Smale horseshoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbits can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed.

8.

Abstract: This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e., not chaotic). Our results show that the transition from regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of “fully developed” chaotic scattering can be reached in our system through a process analogous to the formation of a Smale horseshoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbits can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed.

**1991 **

1. M. Ding, C. Grebogi, E. Ott and J. A. Yorke, Massive bifurcation of chaotic scattering, Phys. Letters 153A (1991), 21-26.

Abstract: In this paper we investigate a new type of bifurcation that occurs in the context of chaotic scattering. The phenomenology of this bifurcation is that the scattering is chaotic on both sides of the bifurcation, but, as the system parameter passes through the critical value, an infinite number of periodic orbits are destroyed and replaced by a new infinite class of periodic orbits. Hence the structure of the chaotic set is fundamentally altered by the bifurcation. The symbolic dynamics before and after the bifurcation, however, remains unchanged.

2. J. A. Kennedy and J. A. Yorke, Basins of Wada, Physica D 51 (l991), 213-225.

Abstract: We describe situations in which there are several regions (more than two) with the Wada property, namely that each point that is on the boundary of one region is on the boundary of all. We argue that such situations arise even in studies of the forced damped pendulum, where it is possible to have three attractor regions coexisting, and the three basins of attraction have the Wada property.

3. H. E. Nusse and J. A. Yorke, Analysis of a procedure for finding numerical trajectories close to chaotic saddle hyperbolic sets, Ergodic Theory and Dyn. Sys., 11 (1991), 189-208.

Abstract: In dynamical systems examples are common in which there are regions containing chaotic sets that are not attractors, e.g. systems with horseshoes have such region. In such dynamical systems one will observe chaotic transients. An important problem is the Dynamical Restraint Problem: given a region that contains a chaotic set but contains no attractor find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time.

We present two procedures (PIM triple procedures) for finding trajectories that stay extremely close to such chaotic sets for arbitrarily long periods of time.

4. B. Hunt and J. A. Yorke, Smooth dynamics on Weierstrass nowhere differentiable curves, Trans. Amer. Math. Soc., 325 (l991), 141-154.

Abstract: We consider a family of smooth maps on an infinite cylinder that have invariant curves that are nowhere smooth. Most points on such a curve are buried deep within its spiked structure, and the outermost exposed points of the curve constitute an invariant subset that we call the facade of the curve. We find that for surprisingly many of the maps in the family, all points in the facades of their invariant curves are eventually periodic.

5. T. Sauer and J. A. Yorke, Rigorous verification of trajectories for the computer simulation of dynamical systems, Nonlinearity 4 (1991), 961-979.

Abstract: We present a new technique for constructing a computer-assisted proof of the reliability of a long computer-generated trajectory of a dynamical system. Auxiliary calculations made along the noise-corrupted computer trajectory determine whether there exists a true trajectory that follow the computed trajectory closely for long times. A major application is to verify trajectories of chaotic differential equations and discrete systems. We apply the main results to computer simulations of the Henon map and the forced damped pendulum.

6. T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Stat. Phys., 65 (1991), 579-616.

Abstract: Mathematical
formulations of the embedding methods commonly used for the reconstruction of
attractors from data series are discussed. Embedding theorems, based on
previous work by H. Whitney and F. Takens, are established for compact subsets
A of Euclidean space Rk. If n is an integer larger than twice the box-counting
dimension of A, then almost every map from R_{k} to R_{n}, in
the sense of prevalence, is one-to-one on A, and moreover is an embedding on
smooth manifolds contained within A. If A is a chaotic attractor of a typical
dynamical system, then the same is true for almost every delay-coordinate map
from R_{k} to R_{n}. These results are extended in two other
directions. Similar results are proved in the more general case of
reconstructions that use moving averages of delay coordinates. Second,
information is given on the self-intersection set that exists when n is less
than or equal to twice the box-counting dimension of A.

7. Z.-P. You, E. J. Kostelich and J. A. Yorke, Calculating stable and unstable manifolds, Int. J. Bifurcation and Chaos 1 (1991), 605-623.

Abstract: A numerical procedure
is described for computing the successive images of a curve under a
diffeomorphism of R^{N}. Given a tolerance g, we show how to rigorously
guarantee that each point on the computed curve lies no further than a distance
g from the true image curve. In particular, if g is the distance between
adjacent points (pixels) on a computer screen, then a plot of the computed
curve coincides with the true curve within the resolution of the display. A
second procedure is described to minimize the amount of computation of parts of
the curve that lie outside a region of interest. We apply the method to compute
the one-dimensional stable and unstable manifolds of the Henon and Ikeda maps,
as well as a Poincare map for the forced damped pendulum.

8. K. Alligood, L. Tedeschini and J. A. Yorke, Metamorphoses: Sudden jumps in basin boundaries, Comm. Math. Phys., 141 (1991), 1-8.

Abstract: In some invertible maps of the plane that depend on a parameter, boundaries of basins of attraction are extremely sensitive to small changes in the parameter. A basin boundary can jump suddenly, and, as it does, change from being smooth to fractal. Such changes are call basin boundary metamorphoses. We prove (under certain non-degeneracy assumptions) that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency.

9. H. E. Nusse and J. A. Yorke, A numerical procedure for finding accessible trajectories on basin boundaries, Nonlinearity, 4 (1991), 1183-1212.

Abstract: In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is non-empty. The basin boundary is either smooth or fractal (that is, it has a Cantor-like structure). When there are horseshoes in the basin boundary, the basin boundary is fractal. A relatively small subset of a fractal basin boundary is said to be accessible from a basin. However, these accessible points play an important role in the dynamics and, especially, in showing how the dynamics change as parameters are varied. The purpose of this paper is to present a numerical procedure that enables us to produce trajectories lying in this accessible set on the basin boundary, and we prove that this procedure is valid in certain hyperbolic systems.

**1992 **

1. I. Kan, H. Kocak and J. A. Yorke, Antimonotonicity: Concurrent creation and annihilation of periodic orbits, Annals of Mathematics 136 (1992), 219-252.

Abstract: One-parameter families
f_{b} of diffeomorphisms of the Euclidean
plane are known to have a complicated bifurcation pattern as b varies near
certain values, namely where homoclinic tangencies are created. We argue that
the bifurcation pattern is much more irregular than previously reported. Our
results contrast with the monotonicity result for the well-understood
one-dimensional family g_{b}(x) = bx(1-x),
where it is known that periodic orbits are created and never annihilated as b
increases. We show that this monotonicity in the creation of periodic orbits
never occurs for any one-parameter family of C^{3} area contracting
diffeomorphisms of the Euclidean plane, excluding certain technical degenerate
cases where our analysis breaks down. It has been shown that in each
neighborhood of a parameter value at which a homoclinic tangency occurs, there
are either infinitely many parameter values at which periodic orbits are
created or infinitely many at which periodic orbits are annihilated. We show
that there are both infinitely many values at which periodic orbits are created
and infinitely many at which periodic orbits are annihilated. We call this
phenomenon antimonotonicity.

2. H. E. Nusse and J. A. Yorke, Border collision bifurcations including period two to period three bifurcation for piecewise smooth systems, Physica D. 57 (1992), 39-57.

Abstract: We examine bifurcation
phenomena for maps that are piecewise smooth and depend continuously on a
parameter m. In the simplest case there is a surface G in phase space along
which the map has no derivative (or has two one-sided derivatives). G is the
border of two regions in which the map is smooth. As the parameter m is varied,
a fixed point Em may collide with the border G, and we may assume that this
collision occurs at m = 0. A variety of bifurcations occur frequently in such
situations, but never or almost never occur in smooth systems. In particular
EmF may cross the border and so will exist for m < 0 and for m > 0 but it
may be a saddle in one case, say m < 0, and it may be a repellor for m >
0. For m < 0 there can be a stable period-two orbit that shrinks to the
point E_{0} as m tends to 0, and for m > 0 there may be a stable
period 3 orbit that similarly shrinks to E_{0} as m tends to 0. Hence
one observes the following stable periodic orbits: a stable period 2 orbit
collapses to a point and is reborn as a stable period 3 orbits. We also see
analogously stable period 2 to stable period p orbit bifurcations, with p =
5,11,52, or period 2 to quasi-periodic or even to a chaotic attractor. We believe
this phenomenon will be seen in many applications.

3. S. P. Dawson, C. Grebogi, J. A. Yorke, I. Kan and H. Kocak, Antimonotonicity: Inevitable reversals of period-doubling cascades, Phys. Letters A 162 (l992), 249-254.

Abstract: In many common nonlinear dynamical systems depending on a parameter, it is shown that periodic orbit creating cascades must be accompanied by periodic orbit annihilating cascades as the parameter is varied. Moreover, reversals from a periodic orbit creating cascade to a periodic orbit annihilating one must occur infinitely often in the vicinity of certain common parameter values. It is also demonstrated that these inevitable reversals are indeed observable in specific chaotic systems.

4. T. Shinbrot, C. Grebogi, J. Wisdom and J. A. Yorke, Chaos in a double pendulum, Am. J. Phys., 60 (1992), 491-499.

Abstract: A novel demonstration of chaos in the double pendulum is discussed. Experiments to evaluate the sensitive dependence on initial conditions of the motion of the double pendulum are described. For typical initial conditions, the proposed experiment exhibits a growth of uncertainties that is exponential with exponent L = 7.5 plus or minus 1.5 s-1. Numerical simulations performed on an idealized model give good agreement, with the value L = 7.9 plus or minus 0.4 s-1. The exponents are positive, as expected for a chaotic system.

5. T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, Using chaos to direct orbits to targets in systems describable by a one-dimensional map, Phys. Rev. A., 45 (l992), 4165-4168.

Abstract: The sensitivity of chaotic systems to small perturbations can be used to rapidly direct orbits to a desired state (the target). We formulate a particularly simple procedure for doing this for cases in that the system is describable by an approximately one-dimensional map, and demonstrate that the procedure is effective even in the presence of noise.

6. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Using chaos to target stationary states of flows, Phys. Letters A 169, (1992), 349-354.

Abstract: The sensitivity of chaotic systems to small perturbations is used to direct trajectories to a small neighborhood of stationary states of three-dimensional chaotic flows. For example, in one of the cases studied, a neighborhood that would typically take 1010 time units to reach without control can be reached using our technique in only about 10 of the same time units.

7. T. Shinbrot, W. Ditto, C. Grebogi, E. Ott, M. Spano and J. A. Yorke, Using the sensitive dependence of chaos (the Butterfly Effect) to direct orbits to targets in an experimental chaotic system, Phys. Rev. Lett. 68 (1992), 2863-2866.

Abstract: In this paper we present the first experimental verification that the sensitivity of a chaotic system to small perturbations (the butterfly effect) can be used to rapidly direct orbits from an arbitrary initial state to an arbitrary accessible desired state.

8. H. E. Nusse and J. A. Yorke, The equality of fractal dimension and uncertainty dimension for certain dynamical systems, Comm. Math. Phys. 150 (1992), 1-21.

Abstract: MGOY introduced the uncertainty dimension as a quantitative measure for final state sensitivity in a system. In MGOY it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.

9. K. T. Alligood and J. A. Yorke, Accessible saddles on fractal basin boundaries, Ergodic Theory and Dyn. Sys. 12 (1992), 377-400.

Abstract: For a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods.

Certain points on the basin boundary are distinguished by being accessible (by a path) from the interior of the basin. For an orientation-preserving homeomorphism, the accessible boundary points have a well-defined rotation number. Under some genericity assumptions, we prove that this rotation number is rational if and only if there are accessible periodic orbits. In particular, if the rotation number is the reduced fraction p/q and if the periodic orbits of periods q and smaller are isolated, then every accessible periodic orbit has minimum period q. In addition, if the periodic orbits are hyperbolic, then every accessible point is on the stable manifold of an accessible periodic point.

10. D. Auerbach, C. Grebogi, E. Ott and J. A. Yorke, Controlling chaos in high dimensional systems, Phys. Rev. Lett. 69 (1992), 3479-3482.

Abstract: Recently formulated techniques for controlling chaotic dynamics face a fundamental problem when the system is high dimensional, and this problem is present even when the chaotic attractor is low dimensional. Here we introduce a procedure for controlling a chaotic time signal of an arbitrarily high dimensional system, without assuming any knowledge of the underlying dynamical equations. Specifically, we formulate a feedback control that requires modeling the local dynamics of only a single or a few of the possible infinite number of phase-space variables.

11. J. A. Alexander, J. A. Yorke, Z-P. You and I. Kan, Riddled Basins, Int. J. Bifurcation & Chaos 2 (1992), 795-813.

Abstract: Theory and examples of attractors with basins that are of positive measure, but contain no open sets, are developed; such basins are called riddled. A theorem is established that states that riddled basins are detected by normal Lyapunov exponents. Several examples, both mathematically rigorous and numerical, motivated by applications in the literature, are presented.

12. B. Hunt, T. Sauer and J. A. Yorke, Prevalence: a translation-invariant "almost every" on infinite dimensional spaces, Bull. Amer. Math. Soc. 27 (1992), 217-238.

Addendum: Bull. Amer. Math. Soc. 28 (1993), 306-307.

Abstract: We present a
measure-theoretic condition for a property to hold almost everywhere on an
infinite-dimensional vector space, with particular emphasis on function spaces
such as C^{k} and L^{p}. Like the concept of Lebesgue almost every
on finite-dimensional spaces, our notion of prevalence is translation
invariant. Instead of using a specific measure on the entire space, we define
prevalence in terms of the class of all probability measures with compact
support. Prevalence is a more appropriate condition than the topological
concepts of open and dense or generic when one desires a probabilistic result
on the likelihood of a given property on a function space. We give several
examples of properties that hold almost everywhere in the sense of prevalence.
For instance, we prove that almost ever C^{1} map on R^{n} has
the property that all of its periodic orbits are hyperbolic.

**1993 **

1. E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Higher dimensional targeting, Phys. Rev. E 47 (1993) 305-310.

Abstract: This paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations. Such a procedure is called targeting. Previous work on targeting for chaotic attractors has been in the context of one- and two-dimensional maps. Here it is shown that targeting can also be done in higher-dimensional cases. The method is demonstrated with a mechanical system described by a four-dimensional mapping whose attractor has two positive Lyapunov exponents and a Lyapunov dimension of 2.8. The target is reached by making very small successive changes in a single control parameter. In one typical case, 35 iterates on average are required to reach a target region of diameter 10-4, as compared to roughly 1011 iterates without the use of the targeting procedure.

2. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Using small perturbations to control chaos, Nature, 363 (1993), pp. 411-417.

Abstract: The extreme sensitivity of chaotic systems to tiny perturbations (the butterfly effect) can be used both to stabilize regular dynamic behaviors and to direct chaotic trajectories rapidly to a desired state. Incorporating chaos deliberately into practical systems therefore offers the possibility of achieving greater flexibility in their performance.

3. M. Ding, C. Grebogi, E. Ott, T. Sauer and J. A. Yorke, Plateau onset for correlation dimension: When does it occur?, Phys. Rev. Lett. 70 (1993), pp. 3872-3873.

Abstract: Chaotic experimental
systems are often investigated using delay coordinates. Estimated values of the
correlation dimension in delay coordinate space typically increase with the
number of delays and eventually reach a plateau (on which the dimension
estimate is relatively constant) whose value is commonly taken as an estimate
of the correlation dimension D_{2} of the underlying chaotic attractor.
We report a rigorous result that implies that, for long enough data sets, the
plateau begins when the number of delay coordinates first exceeds D_{2}.
Numerical experiments are presented. We also discuss how lack of sufficient
data can produce results that seem to be inconsistent with the theoretical
prediction.

4. B. R. Hunt and J. A. Yorke, Maxwell on Chaos, Nonlinear Science Today 3 (1993), pp. 2-4.

5. J.A.C. Gallas, C. Grebogi and J. A. Yorke, Vertices in Parameter Space: Double Crises Which Destroy Chaotic Attractors, Phys. Rev. Lett 71 (1993), pp. 1359-1362.

Abstract:
We report a new phenomenon observed along a crisis locus when two control
parameters of physical models are varied simultaneously: the existence of one
or several vertices. The occurrence of a vertex (loss of differentiability) on
a crisis locus implies the existence of simultaneous sudden changes in the
structure of both the chaotic attractor and of its basin boundary. Vertices
correspond to degenerate tangencies between manifolds of the unstable periodic
orbits accessible from the basin of the chaotic attractor. Physically, small
parameter perturbations (noise) about such vertices induce drastic changes in
the dynamics.

See also J. Milnor, Remarks on iterated
cubic maps, Exp. Math. 1 (1992), 5-24.

6. T. Sauer and J. A. Yorke, How many delay coordinates do you need? Int. J. of Bifurcation and Chaos, 3 (1993) 737-744.

Abstract: Theorems on the use of
delay coordinates for analyzing experimental data are discussed. To reconstruct
a one-to-one correspondence with the state-space attractor, m delay coordinates
are sufficient, where m 2D_{0} (here D_{0} denotes the
box-counting dimension). For calculating the correlation dimension D_{2},
m > D_{2} delays are sufficient. These results remain true under
finite impulse (FIR) filters.

7. Y-C. Lai, C. Grebogi, J. A. Yorke and I. Kan, How often are chaotic saddles nonhyperbolic?, Nonlinearity, 6 (1993), 779-797.

Abstract: In this paper, we numerically investigate the fraction of nonhyperbolic parameter values in chaotic dynamical systems. By a nonhyperbolic parameter value we mean a parameter value at which there are tangencies between some stable and unstable manifolds. The nonhyperbolic parameter values are important because the dynamics in such cases is especially pathological. For example, near each such parameter value, there is another parameter value at which there are infinitely many coexisting attractors. In particular, Newhouse and Robinson proved that the existence of one nonhyperbolic parameter value typically implies the existence of an interval (a Newhouse interval) of nonhyperbolic parameter values. We numerically compute the fraction of nonhyperbolic parameter values for the Henon map in the parameter range where there exist only chaotic saddles (i.e., nonattracting invariant chaotic sets). We discuss a theoretical model that predicts the fraction of nonhyperbolic parameter values for small Jacobians. Two-dimensional diffeomorphisms with similar chaotic saddles may arise in the study of Poincare return map for physical systems. Our results suggest that (1) nonhyperbolic chaotic saddles are common in chaotic dynamical systems; and (2) Newhouse intervals can be quite large in the parameter space.

8. M. Ding, C. Grebogi, E. Ott, T. Sauer and J. A. Yorke, Estimating correlation dimension from a time series: when does plateau onset occur?, Physica D, 69 (1993), 404-424.

Abstract: Suppose that a
dynamical system has a chaotic attractor A with a correlation dimension D_{2}.
A common technique to probe the system is by measuring a single scalar function
of the system state and reconstructing the dynamics in an m - dimensional space
using the delay-coordinate technique. The estimated correlation dimension of
the reconstructed attractor typically increases with m and reaches a plateau
(on which the dimension estimate if relatively constant) for a range of large
enough m values. The plateaued dimension value is then assumed to be an
estimate of D_{2} for the attractor in the original full phase space.
In this paper we first present rigorous results that state that, for a long
enough data string with low enough noise, the plateau onset occurs at m =
Ceil(D_{2}), where Ceil (D_{2}), standing for ceiling of D_{2},
is the smallest integer greater than or equal to D_{2}. We then show
numerical examples illustrating the theoretical prediction. In addition, we
discuss new findings showing how practical factors such as a lack of data and
observational noise can produce results that may seem to be inconsistent with
the theoretically predicted plateau onset at m = Ceil(D_{2}).

9. E. Ott, J. C. Sommerer, J. Alexander, I. Kan and J. A. Yorke, Scaling behavior of chaotic systems with riddled basins, Phys. Rev. Lett., 71 (1993), 4134-4137.

Abstract: Recently it has been shown that there are chaotic attractors whose basins are such that every point in the attractor’s basin has pieces of another attractor's basin arbitrarily nearby (the basin is riddled with holes). Here we report quantitative theoretical results for such basins and compare with numerical experiments on a simple physical model.

10. S. P. Dawson, C. Grebogi, H. Kocak and J. A. Yorke, A geometric mechanism for antimonotonicity in scalar maps with two critical points, Phys. Rev. E 48 (1993), 1676-1682.

Abstract: Concurrent creation and destruction of periodic orbits - antimonotonicity- for one-parameter scalar maps with at least two critical points are investigated. It is observed that if for a parameter value, two critical points lie in an interval that is a chaotic attractor, then, generically, as the parameter is varied through any neighborhood of such a value, periodic orbits should be created and destroyed infinitely often. A general mechanism for this complicated dynamics for one-dimensional multimodal maps is proposed similar to the one of contact-making and contact-breaking homoclinic tangencies in two-dimensional dissipative maps. This subtle phenomenon is demonstrated in a detailed numerical study of a specific one-dimensional cubic map.

11. B. R. Hunt, I. Kan and J. A. Yorke, When Cantor sets intersect thickly, Trans. Amer. Math. Soc., 339 (1993), Number 2, 869-888.

Abstract: The thickness of a Cantor set on the real line is a measurement of its size. Thickness conditions have been used to guarantee that the intersection of two Cantor sets is nonempty. We present sharp conditions on the thicknesses of two Cantor sets that imply that their intersection contains a Cantor set of positive thickness.

**1994 **

1. A. Lasota and J. A. Yorke, Lower bound technique for Markov operators and iterated function systems, Random & Computational Dynamics, 2 (1) (1994), 41-77.

Abstract: A new sufficient condition for asymptotic stability of Markov operators defined on locally compact spaces is proved. This criterion is applied to iterated function systems. In particular it is shown that a nonexpansive iterated function system having an asymptotically stable subsystem is also asymptotically stable.

2. J. A. Kennedy and J. A. Yorke, Pseudocircles in Dynamical systems, Trans. Amer. Math. Soc. (1994), Vol. 343, 349-366.

Abstract: We construct and
example of a smooth map on a 3-manifold that has an invariant set with an
uncountable number of components, countably many of which are pseudocircles. Furthermore,
any map that is sufficiently close (in the C^{1}-metric) to the
constructed map has an invariant set with the same property.

3. H. E. Nusse, E. Ott and J. A. Yorke, Border-Collision Bifurcations: an explanation for observed bifurcation phenomena, Phys. Rev. E, 49 (1994), 1073-1076.

Abstract: Recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena. These phenomena are part of a rich new class of bifurcations that we call border-collision bifurcation. A general criterion for the occurrence of border-collision bifurcations is given. Illustrative numerical results, including transitions to chaotic attractors, are presented. These border-collision bifurcations are found in a variety of physical experiments.

4. E. Ott, J. Alexander, I. Kan, J. Sommerer and J. A. Yorke, Transition to chaotic attractors with riddled basins, Physica D., Vol. 76 (1994), pp. 384-410.

Abstract: Recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby (the basin is riddled with holes). Here we consider the dynamics near the transition to this situation as a parameter is varied. Using a simple analyzable model, we obtain the characteristic behaviors near this transition. Numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered.

5. S. P. Dawson, C. Grebogi, T. Sauer and J. A. Yorke, Obstructions to shadowing when a Lyapunov exponent fluctuates about zero, Phys. Rev. Lett., 73, (1994), pp. 1927-1930.

Abstract: We study the existence or nonexistence of true trajectories of chaotic dynamical systems that lie close to computer-generated trajectories. The nonexistence of such shadowing trajectories is caused by finite-time Lyapunov exponents of the system fluctuating about zero. A dynamical mechanism of the unshadowability is explained through a theoretical model and identified in simulations of a typical physical system. The problem of fluctuating Lyapunov exponents is expected to be common in simulations of higher-dimensional systems.

**1995 **

1. E. Barreto, E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Efficient switching between controlled unstable periodic orbit in higher dimensional chaotic systems, Phys. Rev. E, Vol. 51 (1995), #5, pp. 4169-4172.

Abstract: We develop an efficient targeting technique and demonstrate that when used with an unstable periodic orbit stabilization method, fast and efficient switching between controlled periodic orbits is possible. This technique is particularly relevant to cases of higher attractor dimension. We present a numerical example and report an improvement of up to four orders of magnitude in the switching time over the case with no targeting.

2. A. Pentek, Z. Torozakai, T. Tel, C. Grebogi and J. A. Yorke, Fractal boundaries in open hydrodynamical flows: signatures of chaotic saddles, Phys. Rev. E., Vol. 51 (1995), #5, pp. 4076-4088.

Abstract: We introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently. It is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors. The fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set. We point out that these boundaries coincide with streak lines passing through appropriately chosen points. As an illustrative numerical experiment, we consider a model of the con Karman vortex street, a time periodic two-dimensional flow of a viscous fluid around a cylinder.

3. H. E. Nusse and J. A. Yorke, Border-collision bifurcations for piecewise smooth one-dimensional maps, Int. J. Bifurcation and Chaos, Vol. 5 (1995), No. 1, pp. 189-207.

Abstract: We examine bifurcation
phenomena for one-dimensional maps that are piecewise smooth and depend on a
parameter m. In the simplest case, there is a point c at which the map has no
derivative (it has two one-sided derivatives). The point c is the border of two
intervals in which the map is smooth. As the parameter m is varied, a fixed
point (or periodic point) Em may cross the point c, and we may assume that this
crossing occurs at m = 0. The investigation of what bifurcations occurs at m =
0 reduces to a study of a map fm depending linearly on m and two other
parameters a and b. A variety of bifurcations occur frequently in such
situations. In particular, E_{m} may cross the point c, and for m <
0 there can be a fixed point attractor, and for m > 0 there may be a
period-3 attractor or even a three-piece chaotic attractor which shrinks to E_{0}
as m tends to 0. More generally, for every integer k = 2, bifurcations from a
fixed point attractor to a period-k attractor, a 2k-piece chaotic attractor, a
k-piece chaotic attractor, or a one-piece chaotic attractor can occur for
piecewise smooth one-dimensional maps. These bifurcations are called
border-collision bifurcations. For almost every point in the region of interest
in the (a,b)-space, we state explicitly which border-collision bifurcation
actually does occur. We believe this phenomenon will be seen in many
applications.

4. I. Kan, H. Kocak and J. A. Yorke, Persistent Homoclinic Tangencies in the Henon Family, Physica D, 83 (1995), pp. 313-325.

Abstract: Homoclinic tangencies
in the Henon family fa(x, y) = (a - x_{2} + by, x) for the parameter
values b = 0.3 and a in [1.270, 1.420] are investigated. Our main observation
is that there exist three intervals comprising 93 percent of the values of the
parameter 8 such that for a dense set of parameter values in these intervals
the Henon family possesses a homoclinic tangency. Therefore, one should expect
long parameter intervals where the Henon family is not structurally stable.
Strong numerical support for this observation is provided.

5. J. A. Kennedy and J. A. Yorke, Bizarre Topology is Natural in Dynamical Systems, Bull. Amer. Math. Soc., Vol. 32, #3 (1995), pp. 309-316.

Abstract: We describe an example
of an infinitely differentiable diffeomorphism on a 7-manifold that has a
compact invariant set such that uncountably many of its connected components
are pseudocircles. (Any 7-manifold will suffice.) Furthermore, any
diffeomorphism that is sufficiently close (in the C^{1} metric) to the
constructed map has a similar invariant set, and the dynamics of the map on the
invariant set are chaotic.

6. H. E. Nusse, E. Ott and J. A. Yorke, Saddle-node bifurcations on fractal basin boundaries, Phys. Rev. Lett., 75 (1995). 2482-2485.

Abstract: We demonstrate and analyze a bifurcation producing a type of fractal basin boundary that has the strange property that any point that is on the boundary of that basin is also simultaneously on the boundary of at least two other basins. We give rigorous general criteria guaranteeing this phenomenon, present illustrative numerical examples, and discuss the practical significance of the results.

7. H. B. Stewart, Y. Ueda, C. Grebogi and J. A. Yorke, Double crises in two parameter dynamical systems, Phys. Rev. Lett., 75 (1995). 2478-2481.

Abstract: A crisis is a sudden discontinuous change in a chaotic attractor as a system parameter is varied. We investigate phenomena observed when two parameters of a dissipative system are varied simultaneously, following a crisis along a curve in the parameter plane. Two such curves intersect at a point we call a double crisis vertex. The phenomena we study include the double crisis vertex at which an interior and a boundary crisis coincide, and related forms of double crisis. We show how an experimenter can infer a crisis from observations of other related crises at a vertex.

8. L. Salvino, R. Cawley, C. Grebogi and J. A. Yorke, Predictability in time series, Phys. Letters A, 209 (1995), pp. 327-332.

Abstract: We introduce a technique to characterize and measure predictability in time series. The technique allows one to formulate precisely a notion of the predictable component of given time series. We illustrate our method for both numerical and experimental time series data.

9. C. S. Daw, C.E.A. Finney, M. Vasudevan, N. A. van Goor, K. Nguyen, D. C. Bruns, E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Self organization and chaos in a fluidized bed, Phys. Rev. Lett. (1995), Vol. 75, #12, pp. 2308-2311.

Abstract: We present experimental evidence that a complex system of particles suspended by upward-moving gas can exhibit low-dimensional bulk behavior. Specifically, we describe large-scale collective particle motion referred to as slugging in an industrial device know as a fluidized bed. As gas flow increases from zero, the bulk motion evolves from a fixed point to periodic oscillations to oscillations intermittently punctuated by stutters, which become more frequent as the flow increases further. At the highest flow tested, the behavior become extremely complex (turbulent).

**1996 **

1. H. E. Nusse and J. A. Yorke, Wada basin boundaries and basin cells, Physica D, 90 (1996), pp. 242-261.

Abstract: In dynamical systems examples
are common in which two or more attractors coexist, and in such cases the basin
boundary is nonempty. We consider a two-dimensional diffeomorphism F (that is,
F is an invertible map and both F and its inverse are differentiable with
continuous derivatives), which has at least three basins. Fractal basin
boundaries contain infinitely many periodic points. Generally, only finitely
many of these periodic points are outermost on the basin boundary, that is,
accessible from a basin. For many systems, all accessible points lie on stable
manifolds of periodic points. A point x on the basic boundary is a Wada point
if every open neighborhood of x has a nonempty intersection with at least three
different basins. We call the boundary of a basin a Wada basin boundary if all
its points are Wada points. Our main goal is to have definitions and hypotheses
for Wada basin boundaries that can be verified by computer. The basic notion
basin cell will play a fundamental role in our results for numerical
verifications. Assuming each accessible point on the boundary of a basin B is
on the stable manifold of some periodic orbit, we show that the boundary of the
closure of B is a Wada basin boundary if the unstable manifold of each of its
accessible periodic orbits intersects at least three basins. In addition, we
find condition for basins B_{1}, B_{2},..., B_{N} (N
> 2) under which all B_{i} have the same boundary. Our results
provide numerically verifiable conditions guaranteeing that the boundary of a
basin is a Wada basin boundary. Our examples make use of an existing numerical
procedure for finding the accessible periodic points on the basin boundary and
another procedure for plotting stable and unstable manifolds to verify the
existence of Wada basin boundaries.

2. H. E. Nusse and J. A. Yorke, Basins of attraction, Science (1996), 271, pp. 1376-1380.

Abstract: Many remarkable properties related to chaos have been found in the dynamics of nonlinear physical systems. These properties are often seen in detailed computer studies, but it is almost always impossible to establish these properties rigorously for specific physical systems. This article presents some strange properties about basins of attraction. In particular, a basin of attraction is a Wada basin if every point on the common boundary of that basin and another basin is also on the boundary of a third basin. The occurrence of this strange property can be established precisely because of the concept of a basin cell.

3. A. Lasota and J. A. Yorke, When the long-time behavior is independent of the initial density, SIAM J. of Math. Anal., (1996), Vol. 27, #1, pp. 221-240.

Abstract: This paper investigates dynamical processes for which the state of time t is described by a density function, and specifically dynamical processes for which the shape of the density becomes largely independent of the initial density as time increases. A sufficient condition (weak ergodic theorem) is given for this asymptotic similarity of densities. The processes investigated are in general time dependent, that is, nonhomogeneous in time. Our condition is applied to processes generated by expanding mappings on manifolds, piecewise convex transformations of the unit interval, and integro-differential equations.

4. Y. Lai, C. Grebogi, J. A. Yorke and S. Venkataramani, Riddling bifurcations in chaotic dynamical systems, Phys. Rev. Lett., 77 (1996), pp. 55-58.

Abstract: When a chaotic attractor lies in an invariant subspace, as in systems with symmetry, riddling can occur. Riddling refers to the situation where the basin of a chaotic attractor is riddled with holes that belong to the basin of another attractor. We establish properties of the riddling bifurcation that occurs when an unstable periodic orbit embedded in the chaotic attractor, usually of low period, becomes transversely unstable. An immediate physical consequence of the riddling bifurcation is that an extraordinarily low fraction of the trajectories in the invariant subspace diverge when there is a symmetry breaking.

5. U. Feudel, C. Grebogi, B. Hunt and J. A. Yorke, A map with more than 100 coexisting low-period, periodic attractors, Phys. Rev. E. (1996) 54, pp. 71-81.

Abstract: We study the qualitative behavior of a single mechanical rotor with a small amount of damping. This system may possess an arbitrarily large number of coexisting periodic attractors if the damping is small enough. The large number of stable orbits yields a complex structure of closely interwoven basins of attraction, whose boundaries fill almost the whole state space. Most of the attractors observed have low periods, because high period stable orbits generally have basins too small to be detected. We expect the complexity described here to be even more pronounced for higher-dimensional systems, like the double rotor, for which we find more than 1000 coexisting low-period periodic attractors.

6. E. Kostelich, J. A. Yorke and Z. You, Plotting stable manifolds: error estimates and noninvertible maps, Physica D 93 (1996), pp. 210-222.

Abstract: A numerical procedure
is described that can accurately compute the stable manifold of a saddle fixed
point for a map of R^{2}, even if the map has no inverse. (Conventional
algorithms use the inverse map to compute an approximation of the unstable
manifold of the fixed point.) We rigorously analyze the errors that arise in
the computation and guarantee that they are small. We also argue that a
simpler, nonrigorous algorithm nevertheless produces highly accurate
representations of the stable manifold.

7. B. Peratt and J. A. Yorke, Continuous avalanche mixing of granular solids in a rotating drum, Europhys. Lett. (1996), 35, pp. 31-35.

Abstract: We consider the avalanche mixing of a monodisperse collection of granular solids in a slowly rotating drum. This process has been studied for the case where the drum rotates slowly enough that each avalanche ceases completely before a new one begins (METCALFE G., SHINBROT T., MCCARTHY J. J. and OTTINO J. M., Nature, 374 (1995) 39). We develop a mathematical model for the mixing both in this discrete avalanche case and in the more useful case where the drum is rotated quickly enough to induce a continuous avalanche in the material but slowly enough to avoid significant inertial effects. When applied to the discrete case, our model yields results are consistent with those obtained experimentally by Metcalfe et al.

8. B. Hunt, E. Ott and J. A. Yorke, Fractal dimensions of chaotic saddles of dynamical systems, Phys. Rev. E., (1996), 54, pp. 4819-4823.

Abstract: A formula, applicable to invertible maps of arbitrary dimensionality, is derived for the information dimensions of the natural measures of a nonattracting chaotic set and of its stable and unstable manifolds. The result gives these dimensions in terms of the Lyapunov exponents and the decay time of the associated chaotic transient. As an example, the formula is applied to the physically interesting situation of filtering of data from chaotic systems.

9. J. A. Kennedy and J. A. Yorke, Pseudocircles, diffeomorphisms, and perturbable dynamical systems, Ergodic Theory and Dyn. Sys. (1996), 16, pp. 1031-1057.

Abstract: We construct an
example of a C4 diffeomorphism on a 7-manifold that has an invariant set with
an uncountable number of pseudocircle components. Furthermore, any
diffeomorphism that is sufficiently close (in the C^{1} metric) to the
constructed map has a similar invariant set. We also discuss the topological
nature of the invariant set.

10. D. Auerbach and J. A. Yorke, Controlling chaotic fluctuations in semiconductor laser arrays, J. Optical Soc. Amer. B (1996), Vol. 13, #10, pp. 2178-2187.

Abstract: A control scheme for eliminating the chaotic fluctuations observed in coupled arrays of semiconductor lasers driven high above threshold is introduced. Using the model equations, we show that the output field of the array can be stabilized to a steady in-phase state characterized by a narrow far-field optical beam. Only small local perturbations to the ambient drive current are involved in the control procedure. We carry out a linear stability analysis of the desired synchronized state and find that the number of active unstable modes that are controlled scales with the number of elements in the array. Numerical support for the effectiveness of our proposed control technique in both ring arrays and linear arrays is presented.

11. B. Hunt, K. M. Khanin, Y. G. Sinai and J. A. Yorke, Fractal properties of critical invariant curves, J. Stat. Phys. (1996), Vol. 85, pp. 261-276.

Abstract: We examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers, through both the thermodynamic formalism and numerical computation. The maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond f the curve is destroyed. Our results indicate that the dimension is universal for a given type of singularity and rotation number, and that among all rotation numbers, the golden mean produces the largest dimension.

12. J. C. Alexander, B. Hunt, I. Kan and J. A. Yorke, Intermingled basins for the triangle map, Ergodic Theory and Dyn. Sys. (1996), 16, pp. 651-662.

Abstract: A family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with intermingled basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.

**1997 **

1. M. Sanjuan, J. A. Kennedy, C. Grebogi and J. A. Yorke, Indecomposable continua in dynamical systems with noise: fluid flow past an array of cylinders, Chaos (1997) Vol. 7(1), pp. 125-138.

Abstract: Standard dynamical
systems theory is based on the study of invariant sets. However, when noise is
added, there are no bounded invariant sets. Our goal is then to study the
fractal structure that exists even with noise. The problem we investigate is
fluid flow past an array of cylinders. We study a parameter range for which
there is a periodic oscillation of the fluid, represented by vortices being
shed past each cylinder. Since the motion is periodic in time, we can study a
time-1 Poincare map. Then we add a small amount of noise, so that on each
iteration the Poincare map is perturbed smoothly, but differently for each time
cycle. Fix an x coordinate x_{0} and an initial time t_{0}. We
discuss when the set of initial points at a time t_{0} whose trajectory
(x(t), y(t)) is semibounded (i.e., x(t) > x_{0} for all time) has a
fractal structure called an indecomposable continuum. We believe that the
indecomposable continuum will become a fundamental object in the study of
dynamical systems with noise.

2. B. Hunt, E. Ott and J. A. Yorke, Differentiable generalized synchronism of chaos, Phys. Rev. Lett. E. (1997), Vol. 55, # 4, pp. 4029-4034.

Abstract: We consider simply Lyapunov-exponent based conditions under which the response of a system to a chaotic drive is a smooth function of the drive state. We call this differentiable generalize synchronization (DGS). When DGS does not hold, we quantify the degree of nondifferentiability using the Holder exponent. We also discuss the consequences of DGS and give an illustrative numerical example.

3. H. E. Nusse and J. A. Yorke, The structure of basins of attraction and their trapping regions, Ergodic Theory and Dyn. Sys., (1997), 17, pp. 463-482.

Abstract: In dynamical systems examples are common in which two or more attractors coexist, and in such cases, the basin boundary is nonempty. When there are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the two remaining basins? Is it possible that all three boundaries of these basins coincide? When this last situation occurs the boundaries have a complicated structure. This phenomenon does occur naturally in simply dynamical systems. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. We introduce the basic notion of a basin cell. A basin cell is a trapping region generated by some well-chosen periodic orbit and determines the structure of the corresponding basin. This new notion will play a fundamental role in our main results. We consider diffeomorphisms of a two-dimensional smooth manifold M without boundary, which has at least three basins. A point x in M is a Wada point if every open neighborhood of x has a nonempty intersection with at least three different basins. We call a basin B a “Wada” basin if every x in the boundary of the closure of B is a Wada point. Assuming B is the basin of a basin cell (generated by a periodic orbits P), we show the B is a Wada basin if the unstable manifold of P intersects at least three basins. This result implies conditions for basins B1, B2,..., BN (N>2) to all have exactly the same boundary.

4. E. Barreto, B. Hunt, C. Grebogi, and J. A. Yorke From high dimensional chaos to stable periodic orbits, Phys. Rev. Lett., (1997), Vol. 78, #24, pp. 4561-4564.

Abstract: Regions in the parameter space of chaotic systems that correspond to stable behavior are often referred to as windows. In this Letter, we elucidate the occurrence of such regions in higher dimensional chaotic systems. We describe the fundamental structure of these windows, and also indicate under what circumstances one can expect to find them. These results are applicable to systems that exhibit several positive Lyapunov exponents, and are of importance to both the theoretical and the experimental understanding of dynamical systems.

5. W. Chin, B. Hunt and J. A. Yorke Correlation dimension for iterated function systems, Trans. Amer. Math. Soc. (1997), Vol 349, Number 5, 1783-1796.

Abstract: The correlation dimension of an attractor is a fundamental dynamical invariant that can be computed from a time series. We show that the correlation dimension of the attractor of a class of iterated function systems in RN is typically uniquely determined by the contraction rates of the maps that make up the system. When the contraction rates are uniform in each direction, our results imply that for a corresponding class of deterministic systems the information dimension of the attractor is typically equal to its Lyapunov dimension, as conjectured by Kaplan and Yorke.

6. Z. Toroczkai, G. Karolyi, A. Pentek, T. Tel, C. Grebogi and J. A. Yorke, Wada dye boundaries in open hydrodynamical flows, Physica A., (1997), 239, pp. 235-243.

Abstract: Dyes of different colors advected by two-dimensional flows that are asymptotically simple can form a fractal boundary that coincides with the unstable manifold of a chaotic saddle. We show that such dye boundaries can have the Wada property: every boundary point of a given color on this fractal set is on the boundary of at least two other colors. The condition for this is the nonempty intersection of the stable manifold of the saddle with at least three differently colored domains in the asymptotic inflow region.

7. T. Sauer, C. Grebogi, and J. A. Yorke, How long do numerical chaotic solutions remain valid? Phys. Rev. Lett., (1997), 79, #1, pp. 59-62.

Abstract: We discuss a topological property that we believe provides a useful conceptual characterization of a variety of strange sets occurring in nonlinear dynamics (e.g., strange attractors, fractal basin boundaries, and stable and unstable manifolds of chaotic saddles). Sets with this topological property are known as indecomposable continua. As an example, we give detailed results for the case of an indecomposable continuum that arises from the entrainment of dye advected by a fluid flowing past a cylinder. We show for this case that the indecomposable continuum persists in the presence of small noise.

8. J. A. Kennedy and J. A. Yorke, The topology of stirred fluids, Topology and Its Applications, (1997), 80, pp. 201-238.

Abstract: There are simple idealized mathematical models representing the stirring of fluids. The models we consider involve two fluids entering a chamber, with the overflow leaving it. The stirring created a Cantor-like, but connected, boundary between the fluids that is best-described point-set topologically. We prove that in many cases the boundary between the fluids is an indecomposable continuum.

9. T. Sauer and J. A. Yorke, Are the dimensions of a set and its image equal under typical smooth functions?, Ergodic Theory and Dyn. Sys., (1997), 17, pp. 941-956.

Abstract: We examine the
question whether the dimension D of a set or probability measure is the same as
the dimension of its image under s where s is a typical smooth function, if the
phase space is at least D-dimensional. If m is a Borel probability measure of
bounded support in R^{n} with correlation dimension D, and if k = D,
then under almost every continuously differentiable function (almost every in
the sense of prevalence) from R^{n} to R^{m}, the correlation
dimension of the image of m is also D. If m is the invariant measure of a
dynamical system, the same is true for almost every delay coordinate map, under
weak conditions on periodic orbits. That is, if k = D, the k time delays are
sufficient to find the correlation dimension using a typical measurement
function. Further, it is shown that finite impulse response (FIR) filters do
not change the correlation dimension. Analogous theorems hold for Hausdorff,
pointwise, and information dimension. We show by example that the conclusion
fails for box-counting dimension.

10. M. Sanjuan, J. A. Kennedy, E. Ott and J. A. Yorke, Indecomposable continua and the characterization of strange sets in nonlinear dynamics, Phys. Rev. Lett., (1997), Vol. 78, pp. 1892-1895.

Abstract: We discuss a topological property that we believe provides a useful conceptual characterization of a variety of strange sets occurring in nonlinear dynamics (e.g., strange attractors, fractal basin boundaries, and stable and unstable manifolds of chaotic saddles). Sets with this topological property are known as indecomposable continua. As an example, we give detailed results for the case of an indecomposable continuum that arises from the entrainment of dye advected by a fluid flowing past a cylinder. We show for this case that the indecomposable continuum persists in the presence of small noise.

11. J. Jacobs, E. Ott, T. Antonsen, and J. A. Yorke, Modeling fractal entrainment sets of tracers advected by chaotic temporarily irregular fluid flows using random maps, Physica D110, (1997), 1-17.

Abstract: We model a
two-dimensional open fluid flow that has temporally irregular time dependence
by a random map x_{n+1} = M_{n}(x_{n}), where on each
iterate n, the map Mn is chosen from an ensemble. We show that a tracer
distribution advected through a chaotic region can be entrained on a set that
becomes fractal as time increases. Theoretical and numerical results on the multifractal
dimension spectrum are presented.

12. E. Kostelich, I. Kan, C. Grebogi, E. Ott And J. A. Yorke, Unstable dimension variability: a source of nonhyperbolicity in chaotic systems, Physica D 109 (1997), 81-90.

Abstract: The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In particular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the unstable (and stable) tangent spaces are not constant over the chaotic set; we call this unstable dimension variability. A simple two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed.

**1998 **

1. C. Schroer, T. Sauer, E. Ott and J. A. Yorke, Predicting chaos most of the time from embeddings with self-intersections, Phys. Rev. Lett. (1998), 80, 1410-1413.

Abstract: Embedding techniques for predicting chaotic time series from experimental data may fail if the reconstructed attractor self-intersects, and such intersections often occur unless the embedding dimension exceeds twice the attractor's box counting dimension. Here we consider embedding with self-intersection. When the dimension M of the measurement space exceeds the information dimension D1 of the attractor, reliable prediction is found to be still possible from most orbit points. In particular, the fraction of state space measure from which prediction fails typically scales as epsilon^(M-D1) for small epsilon where epsilon is the diameter of the neighborhood current state used for prediction.

2. U. Feudel, C. Grebogi, L. Poon and J. A. Yorke, Dynamical properties of a simple mechanical system with a large number of coexisting periodic attractors, Chaos, Solitons and Fractals, (1998), Vol. 9, 171-180.

Abstract: We study a simple mechanical system consisting of two rotors that possesses a large number (3000+) of coexisting periodic attractors. A complex fractal boundary separates these tiny islands of stability and their basins of attraction. Hence, the long-term behavior is acutely sensitive to the initial conditions. This sensitivity combined with many periodic sinks give rise to a rich dynamical behavior when the systems is subjected to small amplitude noise. This dynamical behavior is of great utility, and this is demonstrated by using perturbations that are smaller than the noise level to gear and influence the dynamics toward a specific periodic behavior.

3. S. Banerjee, J. A. Yorke and C. Grebogi, Robust chaos, Phys. Rev. Lett. (1998), 80, pp. 3049-3052.

Abstract: Practical applications of chaos require the chaotic orbit to be robust, defined by the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space. We show that robust chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this phenomenon with a practical example from electrical engineering.

4. C. Robert, K. T. Alligood, E. Ott and J. A. Yorke, Outer tangency bifurcations of chaotic sets, Phys. Rev. Lett. (1998), 80, pp. 4867-4870.

Abstract: We present and explain numerical results illustrating the mechanism of a type of discontinuous bifurcation of a chaotic set that occurs in typical dynamical systems. After the bifurcation, the chaotic set acquires new pieces located at a finite distance from its location just before the bifurcation, and these new pieces were not part of a previously existing chaotic set. A scaling law is given describing the creation of unstable periodic orbits following such a bifurcation. We also provide numerical evidence of such a bifurcation for a nonattracting chaotic set of the Henon map.

5. G.-H. Yuan, S. Banerjee, E. Ott and J. A. Yorke, Border-collision bifurcations in the Buck Converter, accepted by IEEE Trans. Circuits and Systems-I: Fund. The. and Appl. (1998), Vol. 45, #7, pp. 707-716..

Abstract: Interesting bifurcation phenomena are observed for the current feedback-controlled buck converter. We demonstrate that most of these bifurcations can be categorized as border-collision bifurcation. A method of predicting the local bifurcation structure through the construction of a normal form is applied. This method applies to many power electronic circuits as well as other piecewise smooth systems.

6. C. Schroer, E. Ott and J. A. Yorke, The effect of noise on nonhyperbolic chaotic attractors, Phys. Re. Let.. (1998), Vol. 81. #7. Pp. 1397-1400.

Abstract: We consider the effect
of small noise of maximum amplitude epsilon on a chaotic system whose noiseless
trajectories limit on a fractal strange attractor. For the case of
nonhyperbolic attractors of two-dimensional maps the effect of noise can be
made much stronger than for hyperbolic attractors. In particular, the maximum
over all noisy orbit point of the distance between the noisy orbit and the
noiseless nonhyperbolic attractor scales like epsilon1/D (D_{1} > 1
is the information dimension of the attractor), rather then like epsilon (the
hyperbolic case). We also find a phase transition in the scaling of the time
averaged moments of the deviations of a noisy orbit from the noiseless
attractor.

7. B. Peratt and J. A. Yorke, Modeling continuous mixing of granular solids in a rotating drum, Physica D 118, (1998), pp. 293-310.

Abstract: We consider the avalanche mixing of a collection of granular solids in a slowly rotating drum. Although not yet well understood, this process has been studied experimentally for the case where the drum rotates slowly enough that each avalanche ceases completely before a new one begins. We develop a mathematical model for the mixing in both the discrete avalanche case and in the more useful case where the drum is rated quickly enough to induce a continuous avalanche in the material but slowly enough to avoid significant inertial effects. This continuous model in turn provides a more plausible model of the discrete avalanche case.

Although avalanches are inherently a nonlinear phenomenon, the mathematical model developed here reduces to a linear integral equation. The asymptotic behavior of the solution for an arbitrary initial distribution is consistent with those obtained experimentally.

8. K. Alligood and J. A. Yorke, Rotation intervals for chaotic sets, Proc. Amer. Math. Soc., (1998), Vol. 126, #9, pp. 2805-2810.

Abstract: Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two-periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.

9. T. Sauer, J. Tempkin and J. A. Yorke, Spurious Lyapunov exponents in attractor reconstruction, Phys. Rev. Lett., (1998), Vol. 81, #20, pp.4341-4344.

Abstract: Lyapunov exponents, perhaps the most informative invariants of a complicated dynamical process, are also among the most difficult to determine from experimental data. In particular, when using embedding theory to build chaotic attractors in a reconstruction space, extra spurious Lyapunov exponents arise that are not Lyapunov exponents of the original system. The origin of these spurious exponents is discussed, and formulas for their determination in the low noise limit are given

10. J. A. Kennedy and J. A.
Yorke, Dynamical system topology preserved in the presence of noise, Turkish J.
Math. Vol. 22 (1998) p. 379.~~ ~~

Abstract: We first give a precise definition of the
terms “topological horseshoe” and “generalized quadrilateral” and then examine
the behavior of a homeomorphism F on a locally compact, separable, locally
connected metric space X (X is usually a manifold in applications) such that F
restricted to some generalized quadrilateral Q in X is a topological horseshoe
map. For a set Q Ì X we define and describe
(1) the “permanent set” Z of Q to be {c Î
X : F^{n}(c)Î
Q for all integers n}, and (2) the “entrainment set” of Q to be E(Q) = {c
Î X : F^{-n}(c)Î
Q for all sufficiently large n}. We give
conditions under which various closed sets of E(Q) are associated, in a strong
way, with indecomposable, closed, connected spaces invariant under F. (A connected set A is indecomposable if it is
not the union of two proper connected sets, each of which is closed relative to
A.) Next we show that even when small
amounts of noise are added to the dynamical system, there are associated
indecomposable sets. These sets are not,
in general, invariant sets for our process with noise, but they are the physically
observable sets, while invariant Cantor sets are not, and they are the sets
that can be measured.

**1999 **

1. B. Hunt, J. Gallas, C. Grebogi, J. A. Yorke and H. Kocak, Bifurcation rigidity, Physica D 129, (1999), pp. 35-56.

Abstract: Bifurcation diagrams of periodic windows of scalar maps are often found to be not only topologically equivalent, but in fact are related by a nearly linear change of parameter coordinates. This effect has been observed numerically for one-parameter families of maps, and we offer an analytical explanation for this phenomenon. We further present numerical evidence of the same phenomenon for two-parameter families, and give a mathematical explanation like that for the one-parameter case.

2. J. A. Kennedy, M.A.F. Sanjuan, J.A. Yorke, and C. Grebogi, The Topology of Fluid Flow Past a Sequence of Cylinders, Topology and Its Applications, 94, (1999), pp. 207-242.

Abstract: This paper analyzes
conditions under which dynamical systems in the plane have indecomposable
continua or even infinite nested families of indecomposable continua. Our
hypotheses are patterned after a numerical study of a fluid flow example, but
should hold in a wide variety of physical processes. The basic fluid flow model
is a differential equation in R^{2}, which is periodic in time, and so
its solutions can be represented by a time-1 map. We represent a version of
this system "with noise" by considering any sequence of maps Fn.

3. D. Sweet, E. Ott and J.A. Yorke, Topology in chaotic scattering, Nature, 399 (May 27, 1999), #6734, pp. 315-316.

The paper has no abstract.

4. T. Sauer and J.A. Yorke, Reconstructing the Jacobian from data with observational noise, Phys. Rev. Lett., 83 (1999), #7, pp. 1331-1334.

Abstract: Methods for the determination of local dynamical linearization information from experimental time series data are subject to computational artifacts. We investigate the artifacts due to observational noise in the data, and give formulas for the expected values of the reconstructed Jacobian in some simple cases. The formulas we derive in the case of realistic noise amplitudes are quite different from those for the noiseless case. In turn, spurious Lyapunov exponents in the noisy case are correspondingly different from the noiseless case.

5. M. Dutta, H.E. Nusse, E. Ott, and J.A. Yorke, Multiple attractor bifurcations: a source of unpredictability in piecewise smooth systems, Phys. Rev. Lett., 83 (1999), pp. 4281-4284.

Abstract: There exists a variety of physically interesting situations described by continuous maps that are nondifferentiable on some surface in phase space. Such systems exhibit novel types of bifurcations in which multiple coexisting attractors can be created simultaneously. The striking feature of these bifurcations is that in the presence of arbitrarily small noise they lead to fundamentally unpredictable behavior of orbits as a system parameter is varied slowly through its bifurcation value. This unpredictability gradually disappears as the speed of variation of the system parameter through the bifurcation is reduced to zero.

6. G.-C. Yuan and J. A. Yorke, An open set of maps for which every point is absolutely nonshadowable, Proc. Amer. Math. Soc., 128 (1999), #3, pp. 909-918.

Abstract: We consider a class of
nonhyperbolic systems, for which there are two fixed points in an attractor
having a dense trajectory; the unstable manifold of one has dimension one and
the other’s is two dimensional. Under the condition that there exists a
direction that is more expanding then other directions, we show that such
attractors are nonshadowable. Using this theorem, we prove that there is an
open set of diffeomorphisms (in the C^{r} – topology, r >1) for
which every point is absolutely nonshadowable, i.e., there exists epsilon >
0 such that, for every delta > 0, almost every delta-pseudo trajectory
starting from this point is epsilon-nonshadowable.

**2000**

1. J. Miller and J.A. Yorke, Finding all periodic orbits of maps using Newton methods: Sizes of basins, PhysicaD 135 (2000), pp. 195-211.

Abstract: For a diffeomorphism F
on R^{2}, it is possible to find periodic orbits of F of period k by
applying ^{k} – I, where I is the identity function. (We
actually use variants of _{k}) for n = 1, 2, 3,…. We show that if p is an
attracting orbit, then there is an open neighborhood of p that is in all the _{k})
for all n. If p is a repelling periodic point of F, it is possible that p is
the only point that is in all of the _{k}) for all n. It is when p is a periodic saddle point of
F that the _{k} (p) and c is
approximately –1 (c ~ -0.84 in Fig. 5). For long periods (k more than about
20), many orbits of F have L so large that the basins are numerically
undetectable. Our main results states that if p is a saddle point of F, the
intersection of _{k}) of p includes a segment of the local stable manifold
of p.

2. G.-C. Yuan and J.A. Yorke, Collapsing of chaos in one dimensional maps, PhysicaD 136 (2000), pp. 18-30.

Abstract: In their numerical investigation of the
family of one dimensional maps ¦ℓ(x) = 1 - 2|x|^{ℓ}, where ℓ
> 2, Diamond et al. [P. Diamond et al., Physica D 86 (1999) 559-571] have
observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random
eventually wind up at -1, a repelling fixed point. This is a numerical artifact because the
continuous maps are chaotic and almost every (true) trajectory can be shown to
be dense in [-1,1]. The goal of this
paper is to extend and resolve this obvious contradiction. We model the numerical simulation with a
randomly selected map. While they used
27 bit precision in computing ¦ℓ, we prove for our model that this numerical
artifact persists for an arbitrary high numerical prevision. The fraction of initial points eventually
winding up at -1 remains bounded away from 0 for every numerical precision.

3. H. E. Nusse and J. A. Yorke, Fractal Basin Boundaries Generated by Basin Cells and the Geometry of Mixing Chaotic Flows, Phys. Rev. Lett., 84 (2000)#4, pp. 626-629.

Abstract: Experiments and computations indicate that mixing in chaotic flows generates certain coherent spatial structures. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifold of some periodic orbit) then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We demonstrate an amazing property for certain global structures: A basin has a basin cell if and only if every diverging curve comes close to every basin boundary point of that basin.

4. S. Banerjee, M.S. Karthik, G.-H. Yuan and J.A. Yorke, Bifurcations in On-Dimensional Piecewise Smooth Maps Theory and Applications in Switching Circuits, IEEE Transactions on Circuits and Systems-I, Vol. 47, #3 (2000) pp. 389-394.

Abstract: The dynamics of a number of switching circuits can be represented by one-dimensional (1-D) piecewise smooth maps under discrete modeling. In this paper we develop the bifurcation theory of such maps and demonstrate the application of the theory in explaining the observed bifurcations in two power electronic circuits.

5. C. Robert, K. Alligood, E. Ott and J.A. Yorke, Explosions of Chaotic Sets, Physica D, 144 (2000), pp. 44-61.

Abstract: Large-scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied. These bifurcations include sudden changes in the size and/or type of the set. An explosion is a bifurcation in which new recurrent points suddenly appear at a non-zero distance from any pre-existing recurrent points. We discuss the following. In a generic on-parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur. (1) a saddle-node bifurcation isolated from other recurrent points, (2) a saddle-node bifurcation embedded in the set of recurrent points, (3) outer homoclinic tangencies, and (4) outer heteroclinic tangencies. (The term “outer tangency” refers to a particular configuration of the stable and unstable manifolds at tangency.) In particular, we examine different types of tangencies of stable and unstable manifolds from orbits of pre-existing invariant sets. This leads to a general theory that unites phenomena such as crises, basin boundary metamorphoses, explosions of chaotic saddles, etc. We illustrate this theory with numerical examples.

6. Y.Z. Xu, Q. Ouyang, J.G. Wu, J.A. Yorke, G.X. Xu, D.F. Xu, R.D. Soloway and J.Q. Ren, Using Fractal to Solve the Multiple Minima Problem in Molecular Mechanics Calculation, Journal of Computational Chemistry, 21, #12 (2000), pp. 1101-1108.

Abstract: This article presents an approach using fractal to solve the multiple minima problem. We use the Newton-Raphson method of the MM3 molecular mechanics program to scan the conformational spaces of a model molecule and a real molecule. The results show each energy minimum, maximum point, and saddle point has a basin of initial points converging to it in conformational spaces. Points converging to different extrema are mixed, and form fractal structures around basin boundaries. Singular points seem to involve in the formation of fractal. When searching within a small region of fractal basin boundaries, the self-similarity of fractal makes it possible to find all energy minima, maxima, and saddle points from which global minimum may be extracted. Compared with other methods, this approach is efficient, accurate, conceptually simple, and easy to implement.

7. S. Guharay, B.R. Hunt, J.A.
Yorke, and O.R. White, Correlations in

Abstract: We report statistical studies of correlation
properties of ~7500 gene sequences, covering coding (exon) and non-coding
(intron) sequences for

8. G.-C. Yuan, J.A. Yorke, T.L. Caroll, E. Ott, L.M. Pecora, Testing whether two chaotic one dimensional processes are dynamically identical, Phys. Rev. Lett 85, (2000), pp. 4265-4268.

Abstract: Consider the situation where two individuals observe the same chaotic physical process but through time series of different measured variables (e.g., one individual measures a temperature and the other measures a voltage). If the two individuals now use their data to reconstruct (e.g., via delay coordinates) a map, the maps they obtain may appear quite different. In the case where the resulting maps appear one dimensional, we introduce a method to test consistency with the hypothesis that they represent the same physical process. We illustrate this method using experimental data from an electric circuit.

**2001**

1. J. A. Kennedy and J.A. Yorke, Topological horseshoes, Trans. Of the Amer. Math. Soc. 353, (2001), #6, pp. 2513-2530.

Abstract: When does a continuous map have chaotic dynamics in a set Q? More specifically, when does it factor over a shift on M symbols? This paper is an attempt to clarify some of the issues when there is no hyperbolicity assumed. We find that the key is to define a “crossing number” for that set Q. If that number is M and M > 1, then Q contains a compact invariant set which factors over a shift on M symbols.

2. D.J. Patil, B.R. Hunt, E. Kalnay, J.A. Yorke, and E. Ott, Local Low Dimensionality of Atmospheric Dynamics, Phys. Rev. Lett. 86, (2001), #26, pp. 5878-5881.

Abstract: A statistic, the BV (bred vector) dimension, is introduced to measure the effective local finite-time dimensionality of a spatiotemporally chaotic system. It is shown that the Earth’s atmosphere often has low BV dimension, and the implications for improving weather forecasting are discussed. (The BV dimension is much lower than the dimension of the attractor.)

3. J. A. Kennedy, S. Kocak and J.A. Yorke, The chaos lemma, The Amer. Math. Monthly, Vol. 108 (2001), #5, pp. 411-423.

Abstract: The complicated behavior of a trajectory of a dynamical system can often be described in terms of its "itinerary", i.e., the order in which it passes through two (or more) sets, say SA and SB. One trajectory's sequence of A's and B's might be (A,B,B,A,A,A,B), while another's might be quite different. In a chaotic system many different sequences are possible. We begin by describing a simple version of the Smale Horseshoe example. There, the two sets are chosen so that every sequence corresponds to some trajectory. We give a new proof that is more elementary than previous proofs and extends easily to new examples of chaotic systems. The proof is based on families of sets that we call "expanders". It has the flavor of proofs for one-dimensional maps, but in those examples, usually only two expander sets are needed, while in our new proof infinitely many expander sets may be used.

4. D. Sweet, H.E. Nusse and J.A. Yorke, Stagger and step method: detecting and computing chaotic saddles in higher dimensions, Phys. Rev. Lett. 86, (2001), #11, PP. 2261-2264.

Abstract: Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices. These transients are caused by the presence of chaotic saddles, and they are a common phenomenon in higher dimensional dynamical systems. For many physical systems, chaotic saddles have a big impact on laboratory measurements, but there has been no way to observe these chaotic saddles directly. We present the first general method to locate and visualize chaotic saddles in higher dimensions.

**2002**

1. C. Grebogi, L. Poon, T. Sauer, J.A. Yorke and D. Auerbach, Shadowability of chaotic dynamical systems, Handbook of Dynamical Systems, 2002, Vol. 2, Ch. 7, pp. 313-344. Edited by B Fiedler. Elsevier Science B. V. (ISBN: 0-444-50168-1)

2. J. A. Tempkin and J. A. Yorke, Measurements of a Physical Process Satisfy a Difference Equation, J. Difference Eq. & Appl., 8 (2002), p. 13-24.

Abstract. To study a physical
process such as a complicated electrical circuit, the investigator measures one
or more physical quantities h at regular time intervals, obtaining a sequence {h_{j} } of measurements. We show that if the
attractor of this physical system has finite box-counting dimension, then for
almost any choice of measurement (in the sense of prevalence), there exists a
continuous scalar difference equation that describes the evolution of the
sequence of measurements.

3. K. Alligood, E. Sander, and J. Yorke, Explosions: global bifurcations at heteroclinic tangencies, Ergodic Theory and Dynamical Systems, Volume 22, (2002), 953-972

Abstract: We investigate bifurcations in the chain recurrent set for a particular class of one-parameter families of diffeomorphisms in the plane. We give necessary and sufficient conditions for a discontinuous change in the chain recurrent set to occur at a point of heteroclinic tangency. These are also necessary and sufficient conditions for an W-explosion to occur at that point.

4. I. Szunyogh, A.V. Zimin, D.J. Patil, B.R. Hunt, E. Kalnay, E. Ott,
and J.A. Yorke, On the Dynamical Basis of Targeting Weather Observations,
Proceedings on Symposium on Observations, Data Assimilation, and Probabilistic
Prediction, Amer. Met. Soc. Jan. 13-17, 2002 Orlando Fl. 197-202.

The paper has no abstract.

**2003**

1. William Ott and James A. Yorke, Learning About Reality From Observation, SIAM Journal on Applied Dynamical Systems, 297-322, Vol. 2, 2003.

Abstract: Takens, Ruelle, Eckmann, Sano and Sawada
launched an investigation of images of attractors of dynamical systems. Let A be a compact invariant set for a map f
on R^{n} and let G map R^{n }to R^{m} where n > m be
a ``typical'' smooth map. When can we
say that A and G (A) are similar, based only on knowledge of the images in Rm
of trajectories in A? For example, under
what conditions on G(A) (and the induced dynamics thereon) are A and G(A)
homeomorphic? Are their Lyapunov
exponents the same? Or, more precisely,
which of their Lyapunov exponents are the same?
This paper addresses these questions with respect to both the general
class of smooth mappings G and the subclass of delay coordinate mappings. In
answering these questions, a fundamental problem arises about an arbitrary
compact set A in R^{n}. For x in
A, what is the smallest integer d such that there is a C^{1} manifold
of dimension d that contains all points of A that lie in some neighborhood of
x? We define a tangent space T_{x}A
in a natural way and show that the answer is: d = dim(T_{x}A). As a
consequence we obtain a Platonic version of the Whitney embedding theorem.

2. M. Corazza, E. Kalnay, D.J. Patil, S.-C. Yang, R. Morss, M.
Cai, I. Szunyogh, B.R. Hunt, and J.A. Yorke,

Use of the Breeding Technique to Estimate the Structure of Analysis "Errors of the Day", Nonlinear Processes in Geophysics, Nonlinear Processes in Geophysics, Vol. 10, pp. 233-243, 2003

3. H. E. Nusse and J. A. Yorke, Characterizing the basins with the most entangled boundaries, Ergodic Theory and Dyn. Sys., 23 (2003). 895-906.

Abstract: In dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is non-empty. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well-chosen periodic orbit), then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We prove the following surprising property for certain fractal basin boundaries: a basin of attraction B has a basin cell if and only if every diverging path in basin B has the entire basin boundary as its limit set. The latter property reflects a complete entangled basin and its boundary.

4. J.A. Kennedy and J.A. Yorke, Generalized Hénon difference equations with delay, Universitatis Iagellonicae Acta Mathematica, XLI (2003), 9-28.

Abstract: Charles Conley once said his goal was to
reveal the discrete in the continuous.
The idea here of using discrete cohomology to elicit the behavior of
continuous dynamical systems was central to his program. We combine this idea with our idea of
"expanders" to investigate a difference equation of the form x_{n}
= F(x_{n-1},…x_{n-m}) when F has a special form. Recall that the equation x_{n} = q(x_{n-1})
is chaotic for continuous real-valued q that satisfies q(0) < 0, q(1/2) >
1, and q(1) < 0. For such a q, it is
also easy to analyze x_{n} = q(x_{n-k}) where k > 1. But when a small perturbation g(x_{n - 1},…x_{n
- m}) is added, the equation

x_{n}
= q(x_{n - k}) + g(x_{n - 1},…x_{n - m})

(where 1 < k < m) is far harder to analyze and appears to require degree theory of some sort. We use k-dimensional cohomology to show that this equation has a 2-shift in the dynamics when g is sufficiently small.

**2004**

**1.** BR Hunt, E. Kalnay, E.J. Kostelich, E. Ott, DJ Patil, T. Sauer,
I. Szunyogh, JA Yorke, and A.V. Zimin,

Four-Dimensional Ensemble Kalman Filtering, Tellus 56A, (2004), 273-277.

Abstract: Ensemble Kalman filtering was developed as a way to assimilate observed data to track the current state in a computational model. In this paper we show that the ensemble approach makes possible an additional benefit; the timing of observations, whether they occur at the assimilation time or at some earlier or later time, can be effectively accounted for at low computational expense. In the case of linear dynamics, the technique is equivalent to instantaneously assimilating data as they are measured. The results of numerical tests of the technique on a simple mode problem are shown.

**2.** M. Brin, *Topology and its
Applications,* **145**, 233-239

Abstract: We study the problem of embedding compact
subsets of *R ^{n}* into

**3.** E. Ott, E., B. R. Hunt, I.
Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, J.
A. Yorke,

A local ensemble Kalman Filter for atmospheric data assimilation.
Tellus 56A (2004), 415-428.

**4.** Ott, E., B. R. Hunt, I. Szunyogh,
A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, and J. A.
Yorke,

Estimating the state of large spatio-temporally chaotic systems,
Phys. Lett. A., 330, (2004) 365-370.

**5.** Michael Roberts, Brian R. Hunt,
and James A. Yorke, Randall Bolanos, and Art Delcher, A Preprocessor for
Shotgun Assembly of Large Genomes, J Comput Biol. 2004;11(4),734-52

Abstract. The whole-genome shotgun (WGS)
assembly technique has been remarkably successful in efforts to determine the
sequence of bases that make up a genome. WGS assembly begins with a large
collection of short fragments that have been selected at random from a genome.
Each of these fragments is then run through a machine which reports the
sequence of bases at each end of the fragment as a sequence of letters called a
``read'', albeit imprecisely. Sequencing errors consist of substitutions,
insertions, and deletions. Each letter in a read is assigned a quality value
that estimates the probability that a sequencing error occurred in determining
that letter. Reads are cut off after about 500 letters, where sequencing errors
become endemic.

We report on a set of procedures that (1) corrects most of
the sequencing errors, (2) changes quality values accordingly, and (3) produces
a list of ``overlaps'', i.e. pairs of reads that plausibly come from
overlapping parts of the genome. Our procedures can be run iteratively and as a
preprocessor for other assemblers. In collaboration with Celera Genomics, we
tested our procedures on their {\it Drosophila} reads. When we replaced
Celera's overlap procedures with ours in the front end of their assembler, it
was able to produce a significantly improved genome.

**6.** Michael Roberts, Wayne Hayes, Brian R. Hunt, Stephen M. Mount,
James A. Yorke, Reducing storage requirements for biological sequence
comparison [Minimizers], Bioinformatics,
Dec 2004; 20: 3363 - 3369.

Abstract:

Motivation: Comparison of
nucleic acid and protein sequences is a fundamental tool of modern
bioinformatics. A dominant method of such string matching is the
"seed-and-extend" approach, in which occurrences of short
subsequences called "seeds" are used to search for potentially longer
matches in a large database of sequences. Each such potential match is then
checked to see if it extends beyond the seed. To be effective, the
seed-and-extend approach needs to catalogue seeds from virtually every
substring in the database of search strings. Projects such as mammalian genome
assemblies and large-scale protein matching, however, have such large sequence
databases that the resulting list of seeds cannot be stored in

Results: We present a simple and elegant method in which only a small fraction of seeds, called "minimizers", needs to be stored. Using minimizers can speed up string-matching computations by a large factor while missing only a small fraction of the matches found using all seeds.

**7.** I. Frommer, E. Harder, B. Hunt, R. Lance, E. Ott and J. Yorke,
Modeling Congested Internet Connections, Proceedings of the IASTED
International Conference on Communications and Computer Networks (Nov 2004),
Cambridge, MA, 2004, 319-324. (This conference referees papers before they are
accepted for presentation.)

Abstract: In this paper, we introduce a continuous-time model aimed at
capturing the dynamics of congested Internet connections. The model combines a
system of differential equations with a sudden change in one of the state
variables. Results from this model show good agreement with the well-known ns
network simulator, better than the results of a previous, similar model. This
is due in large part to the use of the sudden change to reflect the impact of
lost data packets. We also discuss the potential use of this model in network
traffic state estimation.

Proceedings:

T. Sauer, J.A. Yorke, A.V. Zimin, E. Ott, E.J. Kostelich, I. Szunyogh, G. Gyarmati, E. Kalnay, D.J. Patil,

4D Ensemble Kalman Filtering for
Assimilation of Asynchronous Observations, Submitted to

Proceedings:

I. Szunyogh, E.J. Kostelich, G. Gyarmati, B.R. Hunt, E. Ott, A.V. Zimin, E. Kalnay, D.J. Patil, and J.A .Yorke,

A Local Ensemble Kalman Filter for the NCEP GFS Model, Amer. Met. Soc. Proceedings 2004 (a proceedings published on disk so no page numbers exist).

**2005 **

**1. **I. Szunyogh,
E. J. Kostelich, G. Gyarmati, D. J. Patil, B. R. Hunt, E. Kalnay,

E. Ott, and J. A. Yorke,

Assessing a local ensemble
Kalman filter: Perfect model experiments with the NCEP global model. Tellus **57A** (2005) pp 528-545.

Abstract: The accuracy and computational efficiency of the recently proposed
Local Ensemble Kalman Filter (LEKF) data assimilation scheme is investigated on
a state-of-the-art operational numerical weather prediction model using
simulated observations. The model selected for this purpose is the T-62
horizontal- and 28-level vertical-resolution version of the Global Forecast
System (GFS) of the National Centers for Environmental Prediction (NCEP). The
performance of the data assimilation system is assessed for different
configurations of the LEKF scheme.

It is shown that a modest size (40-member) ensemble is sufficient to track the evolution of the atmospheric state with high accuracy. For this ensemble size the computational time per analysis is less than 9 minutes on a cluster of PCs. The analyses are extremely accurate in the mid-latitude storm track regions. The largest analysis errors, which are typically much smaller than the observational errors, occur where parameterized physical processes play important roles. Since these are also the regions where model errors are expected to be the largest, limitations of a real-data implementation of the ensemble based Kalman filter may be easily mistaken for model errors. In light of these results, the importance of testing the ensemble based Kalman filter data assimilation systems on simulated observations is stressed.

**2.** Brandy L. Rapatski, Frederick
Suppe, and James A. Yorke, HIV Epidemics Driven by Late Disease-Stage
Transmission, JAIDS, Journal of Acquired Immune Deficiency Syndromes, 38, 2005,
241-253.

Abstract: How infectious a person is when infected with HIV depends upon what stage of the disease the person is in. We use three stages, which we call primary, asymptomatic and symptomatic. It is important to have a systematic method for computing all three infectivities so that the measurements are comparable. Using robust modeling we provide high-resolution estimates of semen infectivity by HIV disease stage. We find that the infectivity of the symptomatic stage is far higher, hence more potent, than the values that prior studies have used when modeling HIV transmission dynamics. The stage infectivity rates for semen are 0.024, 0.002, 0.299 for primary, asymptomatic and symptomatic (late-stage) respectively. Implications of our infectivity estimates and modeling for understanding heterosexual epidemics such as the Sub-Saharan African one are explored.

**3.** John Harlim, Mike Oczkowski, James A. Yorke, Eugenia Kalnay, and
Brian R. Hunt, Convex Error Growth Patterns in a Global Weather Model, Phys.
Rev. Lett. **94** (2005), 228501:1-4.

Abstract: We investigate the
error growth, that is, the growth in the distance E between two typical
solutions of a weather model. Typically E grows until it reaches a saturation
value E_{s}. We find two distinct broad log-linear regimes, one for E
below 2% of E_{s} and the other for E above. In each, log(E=E_{s})
grows as if satisfying a linear differential equation. When plotting d
log(E)=dt vs log(E), the graph is convex. We argue this behavior is quite
different from other dynamics problems with saturation values which yield
concave graphs.

**4.** William Ott and J. A. Yorke, Prevalence, Bull. Amer. Math. Soc.
42 (2005), 263-290.

Abstract. Many problems in mathematics and science require the use of infinite-dimensional spaces. Consequently, there is need for an analogue of the finite-dimensional notions of “Lebesgue almost every” and “Lebesgue measure zero” in the infinite-dimensional setting. The theory of prevalence addresses this need and provides a powerful framework for describing generic behavior in a probabilistic way. We survey the theory and applications of prevalence.

**5.** Steven L. Salzberg and James A. Yorke, Beware of mis-assembled
genomes, Letter to the editor, (2005) V 21 (no. 24): 4320-4321 Bioinformatics.

**6.** R. Lance, I. Frommer, B. R. Hunt, E. Ott, J. A. Yorke, E.
Harder,

Round-trip time inference via passive monitoring,

Proc. of the Workshop on Large Scale Network Inference (LSNI):

Methods, Validation, and Applications, ACM SIGMETRICS (June 2005, Banff, Alberta,
Canada).

**7. **J. Yorke, Chaos, New Scientist, 187 (

**2006**

**1. **Brandy L. Rapatski, Frederick Suppe, and James A. Yorke,
Reconciling different infectivity estimates for HIV-1, JAIDS, Journal of Acquired
Immune Deficiency Syndromes,** **Volume 43(3) 1 November 2006 pp 253-256.

**2. **C.M. Danforth, J.A. Yorke, Making Forecasts for Chaotic Physical
Processes, Physical Review Letters, 96, 144102 (2006).

Abstract.
Making a prediction for a chaotic physical process involves specifying the
probability associated with each possible outcome. Ensembles of solutions are
frequently used to estimate this probability distribution. However, for a
typical chaotic physical system *H *and model *L *of that system, no
solution of *L *remains close to *H *for all time. We propose an
alternative. This Letter shows how to inflate or systematically perturb the
ensemble of solutions of *L *so that some ensemble member remains close to
*H *for orders of magnitude longer than unperturbed solutions of *L*.
This is true even when the perturbations are significantly smaller than the
model error.

**3. **Joseph D. Skufca, James A. Yorke, and Bruno Eckhardt, The edge
of chaos in a parallel shear flow, *96*

**Abstract.** We study the transition between laminar and turbulent
states in a Galerkin representation of a parallel shear flow, where a stable
laminar flow and a transient turbulent flow state coexist. The regions in
initial conditions where the lifetimes show strong fluctuations and a sensitive
dependence on initial conditions are separated from the ones with a smooth
variation of lifetimes by the edge of chaos. We describe a technique to
identify and follow the edge of chaos and provide evidence that it is a
surface. For low Reynolds numbers we find that the surface that is the edge of
chaos coincides with the stable manifold of a periodic orbit, whereas at higher
Reynolds numbers it is the stable set of a higher-dimensional chaotic object.

**4. **K.T. Alligood, E. Sander, and J.A. Yorke, Three-dimensional
crisis: Crossing bifurcations and unstable dimension variability. Phys. Rev.
Lett. 96 (2006), 244103.

**Abstract:** A crisis is a global bifurcation in which a chaotic
attractor has a discontinuous change in size or suddenly disappears as a scalar
parameter of the system is varied. Examples of crises for two dimensional maps
occur simultaneously with tangencies of stable and unstable manifolds of under
underlying saddle orbits. Here we describe a different type of global
bifurcation, which we call a “crossing" bifurcation, which can result in a
crisis. This bifurcation does not involve a tangency and cannot occur in maps
of dimension smaller than three. An important distinction in the type of global
bifurcation is made, depending on whether the crossing invariant manifolds are
twisted or not. We introduce this new concept by presenting an example of a
parametrized three dimensional chaotic attractor which undergoes a crisis at a
crossing bifurcation with twisted manifolds. The crisis also produces unstable
dimension variability in the system.

**2007**

**1.** K.T. Alligood,
E. Sander, J.A. Yorke, Explosions in
dimensions one through three. Rend. Sem. Mat. Univ. Pol. Torino - Vol.
65, 1 (2007), pp 1-15. This special issue was entitled “Subalpine Rhapsody in
Dynamics”

Abstract. Crises are discontinuous changes in the size of a chaotic attractor as a parameter is varied. A special type of crisis is an explosion, in which the new points of the attractor form far from any previously recurrent points. This article summarizes new results in explosions in dimension one, and surveys previous results in dimensions two and three. Explosions can be the result of homoclinic and heteroclinic bifurcations. In dimensions one and two, homoclinic and heteroclinic bifurcations occur at tangencies. We give a classification of one-dimensional explosions through homoclinic tangency. We describe our previous work on the classification of planar explosions through heteroclinic tangencies. Three-dimensional heteroclinic bifurcations can occur without tangencies. We describe our previous work, which gives an example of such a bifurcation and explains why three-dimensional crossing bifurcations exhibit unstable dimension variability, a type of non-hyperbolic behavior which results in a breakdown of shadowing. In addition, we give details for a new scaling law for the parameter-dependent variation of the density of the new part of the chaotic attractor.

**2. **Tobias M. Schneider, James A. Yorke,
and Bruno Eckhardt, Turbulence Transition and the Edge of Chaos in Pipe Flow,
20 July 2007 Phys. Rev. Lett. (Vol.99, No.3):

Abstract. The linear stability of pipe flow implies that sufficiently only perturbations of sufficient strength will trigger turbulence. In order to determine this threshold we study the edge of chaos which separates perturbations that decay towards the laminar profile and perturbations that trigger turbulence. Using the lifetime as an indicator and methods developed in (Skufca et al, Phys. Rev. Lett. 96, 174101 (2006)) we show that superimposed on an overall 1=Re-scaling predicted and studied previously there are small, non-monotonic variations reflecting folds in the basin boundary. We also trace the motion in the edge and find that it is formed by the stable manifold of flow fields that are dominated by pairs of downstream vortices, asymmetrically placed towards the wall.

**3. **Aleksey V Zimin, Douglas R. Smith, Granger Sutton, James A.
Yorke, Assembly Reconciliation, Bioinformatics 24 (2007) 42-45.

**Abstract.**

Motivation: Many genomes are sequenced by a collaboration of several centers, and then each center produces an assembly using their own assembly software. The collaborators then pick the draft assembly that they judge to be the best and the information contained in the other assemblies is usually not used.

Methods: We have developed a technique that we call assembly reconciliation that can merge draft genome assemblies. It takes one draft assembly, detects apparent errors, and, when possible, patches the problem areas using pieces from alternative draft assemblies. It also closes gaps in places where one of the alternative assemblies has spanned the gap correctly.

Results: Using the Assembly Reconciliation technique we produced reconciled assemblies of six Drosophila species in collaboration with Agencourt Bioscience and The J. Craig Venter Institute. These assemblies are now the official (CAF1) assemblies used for analysis. We also produced a reconciled assembly of Rhesus Macaque genome, and this assembly is available from our website http://www.genome.umd.edu.

Availability:
The reconciliation software is available for download from http://www.genome.umd.edu/software.htm.^{
}

**4. **Joshua A. Tempkin & J. A. Yorke, Spurious Lyapunov Exponents
Computed from Data, SIAM J. Appl. Dyn. Syst. (SIADS) **6**,
457-474 (2007)

**Abstract.** Lyapunov exponents
can be difficult to determine from experimental data. In particular, when using
embedding theory to build chaotic attractors in a reconstruction space, extra
\spurious" Lyapunov exponents can arise that are not Lyapunov exponents of
the original system. By studying the local linearization matrices that are key
to a popular method for computing Lyapunov exponents, we determine explicit
formulas for the spurious exponents in certain cases. Notably, when a
two-dimensional system with Lyapunov exponents α and β is reconstructed
in a five-dimensional space by a generic embedding, the reconstructed system
has exponents α, β, 2 α, α + β, and 2β.

**5. **J. Kennedy and J.A. Yorke, Shadowing in Higher Dimensions, *Differential Equations, Chaos and
Variational Problems*, V. Staicu (Ed.), Birkhäuser, pp.
241-246, 2008.

**6.** ** **Drosophila 12 Genome Consortium, 450
authors including A. Zimin and J.A. Yorke, Evolution of genes and genomes on
the *Drosophila* phylogeny,

**Nature** 203-218, Vol 450,

**7. **Helena E.
Nusse and J. A. Yorke, Bifurcations of attraction from the view point of prime
ends, Topology and its Applications Volume 154,
Issue 13, 1 July 2007, Pages 2567-2579, The Proceedings of the
US–Polish International Workshop on Geometric Methods in Dynamical Systems

Abstract. In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty and the basins often have fractal basin boundaries. The purpose of this paper is to describe the structure and properties of unbounded basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Caratheodory, Freudenthal and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. In all well known examples, most points are of type 1. Many two-dimensional basins have a basin cell, that is, a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well chosen periodic orbit. Then the basin consists of a central body (the basin cell) and a finite number of channels attached to it, and the basin boundary is fractal. We present a result that says {a basin has a basin cell} if and only if {every prime end that is defined by a chain of unbounded regions (in the basin) is a prime end of type 3 and furthermore all other prime ends are of type 1}. We also prove as a parameter is varied, the basin cell for a basin B is created (or destroyed) if and only if either there is a saddle node bifurcation or the basin B has a prime end that is defined by a chain of unbounded regions and is a prime end of either type 2 or type 4.

**8. **J. Kennedy, D.R. Stockman, J.A. Yorke,

Inverse limits and an implicitly defined difference equation from economics,

Topology and its applications, 154 (2007), 2533-2553.

**9.** David D. Kuhl, Istvan Szunyogh, Eric J. Kostelich, D. J. Patil, Gyorgyi
Gyarmati, Michael Oczkowski, Brian R. Hunt, Eugenia Kalnay, Edward Ott, and
James A. Yorke,

Assessing Predictability with a
Local Ensemble Kalman Filter, *Journal of the Atmospheric Sciences*, Vol.
64 (2007), No. 4, pages 1116–1140.

Abstract. In this paper, the spatio-temporally changing nature of predictability is studied in the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS), a state-of-the-art numerical weather prediction model. Atmospheric predictability is assessed in the perfect-model scenario for which forecast uncertainties are entirely due to uncertainties in the estimates of the initial states. Initial conditions (analyses) are obtained by assimilating simulated noisy observations of the \true" states with the Local Ensemble Kalman Filter (LEKF) data assimilation scheme. For this specific choice of the model and data assimilation system, the forecast errors grow exponentially in the extra-tropics and linearly in the tropics. The analysis errors are the smallest in the regions, the extratropical storm tracks, where the growth of the forecast errors is the fastest. This seemingly paradoxical result is due to the strong anti-correlation between the local dimensionality and the error variance explained by the LEKF ensemble. This strong anti-correlation makes the LEKF algorithm extremely efficient in estimating the analysis and forecast uncertainties in the regions of local low dimensionality, which coincide with the regions fastest error growth. The efficient estimation of the space of uncertainties enables the LEKF to produce very accurate analyses and very accurate estimates of the forecast uncertainties. It is conjectured that the results presented here could be reproduced with other suitably formulated ensemble-based Kalman Filter data assimilation schemes.

**Proceedings.**

Szunyogh, I., E. A. Satterfield, J. A. Aravequia, E. J. Fertig, G. Gyarmati, E. Kalnay, B. R. Hunt, E. J. Kostelich, D. D. Kuhl, E. Ott, and J. A. Yorke,

The Local Ensemble Transform Kalman Filter and its implementation on the NCEP global model at the University of Maryland.

Workshop
Proceedings, Flow Dependent Aspects of Data Assimilation,

**2008**

**1. **I. Szunyogh, E.J. Kostelich, G.
Gyarmati, E. Kalnay, B. R. Hunt, E. Ott, E. Satterfield, J.A. Yorke,

A local ensemble transform Kalman filter data assimilation system for the NCEP global model, Tellus (2008), 60A, 113–130.

Abstract. The accuracy and computational efficiency of a parallel computer implementation of the Local Ensemble Transform Kalman Filter (LETKF) data assimilation scheme on the model component of the 2004 version of the Global Forecast System (GFS) of the National Centers for Environmental Prediction (NCEP) is investigated. Numerical experiments are carried out at model resolution T62L28. All atmospheric observations that were operationally assimilated by NCEP in 2004, except for satellite radiances, are assimilated with the LETKF. The accuracy of the LETKF analyses is evaluated by comparing it to that of the Spectral Statistical Interpolation (SSI), which was the operational global data assimilation scheme of NCEP in 2004. For the selected set of observations, the LETKF analyses are more accurate than the SSI analyses in the Southern Hemisphere extratropics and are comparably accurate in the Northern Hemisphere extratropics and in the Tropics. The computational wall-clock times achieved on a Beowulf cluster of 3.6 GHz Xeon processors make our implementation of the LETKF on the NCEP GFS a widely applicable analysis-forecast system, especially for research purposes. For instance, the generation of four daily analyses at the resolution of the NCAR-NCEP reanalysis (T62L28) for a full season (90 d), using 40 processors, takes less than 4 d of wall-clock time.

**2. **Michael Roberts, Aleksey V. Zimin, Wayne Hayes, Brian R. Hunt,
Cevat Ustun, James R. White, Paul Havlak, and James Yorke,

Improving Phrap-based Assembly of the Rat
Using “Reliable” Overlaps,

PLoS

Published online 2008 March 19. doi: 10.1371/journal.pone.0001836

**Abstract.
**The assembly methods used for whole-genome shotgun (WGS) data have a major
impact on the quality of resulting draft genomes. We present a novel algorithm
to generate a set of “reliable” overlaps based on identifying repeat k-mers. To
demonstrate the benefits of using reliable overlaps, we have created a version
of the Phrap assembly program that uses only overlaps from a specific list. We
call this version *PhrapUMD*. Integrating PhrapUMD and our “reliable-overlap”
algorithm with the Baylor College of Medicine assembler, Atlas, we assemble the
BACs from the *Rattus norvegicus* genome project. Starting with the same
data as the Nov. 2002 Atlas assembly, we compare our results and the Atlas
assembly to the 4.3 Mb of rat sequence in the 21 BACs that have been finished.
Our version of the draft assembly of the 21 BACs increases the coverage of
finished sequence from 93.4% to 96.3%, while simultaneously reducing the base
error rate from 4.5 to 1.1 errors per 10,000 bases. There are a number of ways
of assessing the relative merits of assemblies when the finished sequence is
available. If one views the overall quality of an assembly as proportional to
the inverse of the product of the error rate and sequence missed, then the
assembly presented here is seven times better. The UMD Overlapper with options
for reliable overlaps is available from the authors at http://www.genome.umd.edu.
We also provide the changes to the Phrap source code enabling it to use only
the reliable overlaps.

**3. **Samuel Zambrano, Miguel A. F.
Sanju´an, James A. Yorke, Partial
control of chaotic systems, a Rapid Communication in Physical Review E, Phys.
Rev. E 77, 055201 (2008).

**Abstract. **In a region in phase space
where there is a chaotic saddle, all initial conditions will escape from it
after a transient with the exception of a set of points of zero Lebesgue
measure. The action of an external noise makes all trajectories escape faster.
Attempting to avoid those escapes by applying a control smaller than noise
seems to be an impossible task. Here we show however that this goal is indeed
possible, based on a geometrical property found typically in this situation,
the existence of a horseshoe. The horseshoe implies that there exists what we
call safe sets, which assures that there is a general strategy that allows us
to keep trajectories inside that region with control smaller than noise. We
call this type of control “partial control of chaos”.

**4. **William Ott and James A. Yorke, When Lyapunov exponents fail to
exist,

Phys. Rev. E **78**,
056203 (2008) [6 pages]

**5.** Suzanne S. Sindi, Brian R. Hunt, and
James A. Yorke,

Duplication
Count Distributions in

Phys. Rev. E, December 2008 issue (Vol.78, No.6), 061912, [11 pages]

Abstract. We study quantitative features of complex repetitive sequences in several genomes; these sequences are sufficiently long that they are unlikely to have repeated by chance. For each genome we study, we determine the number of identical copies, the “duplication count”, of each sequence of length 40, that is of each “40-mer”. We say a 40-mer is “repeated” if its duplication count is at least 2. We focus mainly on “complex” 40-mers, those without short internal repetitions. We find that we can classify most of the complex repeated 40-mers into two categories: one category has its copies clustered closely together on one chromosome, the other has its copies distributed widely across multiple chromosomes. For each genome and each of the categories above, we compute N(c), the number of 40-mers that have duplication count c, for each integer c. In each case, we observe a power-law-like decay in N(c) as c increases from 3 to 50 or higher. In particular, N(c) decays much more slowly than would be predicted by evolutionary models where each 40-mer is equally likely to be duplicated. We discuss evolutionary models that do reflect the slow decay of N(c).

**6. **J. A. Kennedy,
D. R. Stockman, J. A. Yorke, Inverse Limits and Models with Ill-Defined Forward
Dynamics, J. Math. Economics, 44(2008), 423-444.

Abstract. Some economic models like the cash-in-advance model of money have the property that the dynamics are ill-defined going forward in time, but well-defined going backward time. In this paper, we apply the theory of inverse limits to characterize topologically all possible solutions to a dynamic economic model with this property. We show that such techniques are particularly well suited for analyzing the dynamics going forward in time even though the dynamics are ill defined in this direction. In particular, we analyze the inverse limit of the cash-in-advance model of money and illustrate how information about the inverse limit is useful for detecting or ruling out complex dynamics.

**7**. James R.** **White; M. Roberts; JA Yorke, M. Pop,

Figaro: a novel statistical method for vector sequence removal,

Bioinformatics 24 (2008) 462-467.

**8. **Szunyogh, I,
E. J. Kostelich, G. Gyarmati, E. Kalnay, B. R. Hunt, E. Ott, Elizabeth
Satterfield, and J. A. Yorke,

A local ensemble transform Kalman filter data assimilation system for the
NCEP global model, Tellus Series A – Dynamic Met. and Oceanography, 60 (2008)
113-130.

**2009**

**1. **Russell D.
Halper, Eric J. Harder, Brian R. Hunt, James A. Yorke,

Stability of

**Abstract.** We study
the stability of the dynamics of a deterministic model of

**2. **Aleksey V.
Zimin, Arthur L. Delcher, Liliana
Florea, David A. Kelley, Finian Hanrahan, Guillaume Marcais, Geo Pertea,
Daniela Puiu, Michael Roberts, Michael C. Schatz, Poorani Subramanian, Curt Van
Tassell, James A. Yorke, and Steven L. Salzberg,

A whole-genome assembly of the domestic cow, Bos taurus,

*Genome Biology* 2009,

**Abstract. **The
genome of the domestic cow, Bos taurus, was sequenced
using a mixture of hierarchical and whole-genome shotgun sequencing
methods. We have assembled the 35
million raw sequence reads and applied a variety of assembly improvement
techniques, creating an assembly of 2.86 billion base pairs that has multiple
improvements over previous assemblies: (1) the new assembly is more complete,
covering more of the genome; (2) thousands of gaps have been closed; (3) many
erroneous inversions, deletions, and translocations have been corrected; and
(3) thousands of single-nucleotide errors have been corrected. Our evaluation using independent metrics
demonstrates that the resulting assembly is substantially more accurate and
complete than alternative versions. By
using independent mapping data and conserved synteny between the cow and human
genomes, we were able to construct an assembly with excellent large-scale
contiguity in which a large majority (~91%) of the genome has been placed onto
the 30 Bos taurus chromosomes. We constructed a new cow-human synteny map
that expands upon previous maps. We also
identify for the first time a portion of the Bos taurus
Y chromosome.

**3. **E. Sander
& J.A. Yorke, A classification of explosions in dimension
one.

Ergodic Theory and Dynamical Systems, 29 (2009), 715-731. http://arxiv.org/abs/math/0703176

Abstract: A discontinuous change in the size of an attractor is the most easily
observed type of global bifurcation. More generally, an explosion is a
discontinuous change in the set of recurrent points. An
explosion often results from heteroclinic and homoclinic tangency bifurcations.
We prove that for one-dimensional maps planar explosions are
generically the result of either tangency or saddle-node
bifurcations. Furthermore, we give necessary and sufficient
for generic tangency bifurcations to lead to explosions.

**4. **Brandy Rapatski & James Yorke,
Modeling HIV outbreaks: The male to female prevalence ratio in the core
population, Mathematical Biosciences and Engineering, MBE 6 (2009),
135-143.

**Abstract. **What affects the male-female
infected ratio in the core heterosexual population in an HIV epidemic? Hethcote
& Yorke [1] introduced the term “core” initially to loosely describe the
collection of individuals having the most unprotected sex partners. We study
the early epidemic during the exponential growth phase and focus on the core
group where most of the HIV transmissions occur. We argue that in the early
outbreak phase of an epidemic, there is an identity which we call the outbreak
equation. It relates three ratios that describe the core men versus the core
women, namely, the ratio *E *of numbers of core men to core women, the
ratio *C *of numbers of infected core men to core women, and the ratio *M
*of the infectiousness of core men to core women. Then the relationship
between the ratios is *E *= *MC*^{2} in the early outbreak
phase. We investigate two very different scenarios, one in which there are two
times as many core men as core women (*E *= 2) and the other in which core
men equal core women (*E *= 1). In the first case, the HIV epidemic grows
at a much faster rate. We conclude that if the female core group was larger,
that is, women in the population were more promiscuous (or if men were less
promiscuous) then the HIV epidemic would grow slower.

**5. **Ian Frommer, Eric Harder, Brian
Hunt, Ryan Lance, Edward Ott, James Yorke,

Bifurcation and chaos in a
periodically probed computer network,

International Journal of
Bifurcation and Chaos, Vol.
19, No. 9 (2009) 3129–3141

**6. **Evelyn Sander and James A. Yorke,

Period-doubling
cascades for large perturbations of Henon families,

Journal
of Fixed Point Theory and Applications, 6(1): 153-163, 2009, DOI:
10.1007/s11784-009-0116-7

http://arxiv.org/abs/0903.3607

**Abstract **The Henon family has been shown to have period-doubling cascades.
We show here that the same occurs for a much larger class: Large perturbations
do not destroy cascades. Furthermore, we can classify the period of a cascade
in terms of the set of orbits it contains, and count the number of cascades of
each period. This class of families extends a general theory explaining why
cascades occur.

**2010**

**1.** **Turkey** paper:

Rami Dalloul, Julie Long, **Aleksey Zimin**, Luqman
Aslam, Kathryn Beal, Le Ann Blomberg,
David Burt, Oswald Crasta, Richard Crooijmans, Kristal Cooper, Roger
Coulombe, Supriyo De, Mary Delany, Jerry Dodgson, Jennifer
Dong, Clive Evans, Paul Flicek, Liliana Florea, Otto Folkerts, Martien Groenen, Tim Harkins, Javier Herrero,
Steve Hoffmann, Hendrick-Jan Megens,
Andrew Jiang, Pieter de Jong, Pete Kaiser, Heebal
Kim, Kyu-Won Kim, Sungwon
Kim, David Langenberger, Mi-Kyung Lee, TaeHeon Lee, Shrinivasrao Mane,
Guillaume Marcais, Manja Marz,
Audrey McElroy, Thero Modise,
Mikhail Nefedov, Cedric Notredame,
Ian Paton, William Payne, Geo Pertea, Dennis Prickett, Daniela Puiu, Dan Qioa,
Emanuele Raineri, Magali Ruffier, Steven Salzberg,
Michael Schatz, Chantel Scheuring,
Carl Schmidt, Steven Schroeder, Stephen Searle, Edward Smith, Jacqueline Smith,
Tad Sonstegard, Peter Stadler,
Hakim Tafer, Zhijian Tu,
Curtis Van Tassell, Albert Vilella, Kelly Williams, **James Yorke**, Liqing
Zhang, Hong-Bin Zhang, Xiaojun Zhang, Yang Zhang, and
Kent Reed;

**Multi-platform Next
Generation Sequencing of the Domestic Turkey ( Meleagris
gallopavo): Genome Assembly and Analysis, **

PLoS Biology. Published Sept 7 2010

**Abstract.** A synergistic combination of
two next-generation sequencing platforms with a detailed comparative BAC
physical contig map provided a cost-effective assembly of the genome sequence
of the domestic turkey (Meleagris gallopavo).
Heterozygosity of the sequenced source genome allowed discovery of more than
600,000 high quality single nucleotide variants. Despite this heterozygosity,
the current genome assembly (~1.1 Gb) includes 917 Mb
of sequence assigned to specific turkey chromosomes. Annotation identified
nearly 16,000 genes, with 15,093 recognized as protein coding and 611 as
non-coding RNA genes. Comparative analysis of the turkey, chicken, and zebra
finch genomes, and comparing avian to mammalian species, supports the
characteristic stability of avian genomes and identifies genes unique to the
avian lineage. Clear differences are seen in number and variety of genes of the
avian immune system where expansions and novel genes are less frequent than
examples of gene loss. The turkey genome sequence provides resources to further
understand the evolution of vertebrate genomes and genetic variation underlying
economically important quantitative traits in poultry. This integrated approach
may be a model for providing both gene and chromosome level assemblies of other
species with agricultural, ecological, and evolutionary interest.

**2. **Madhura R. Joglekar, Evelyn Sander,
and James A. Yorke,

Fixed points indices and period-doubling cascades. Journal
of Fixed Point Theory and Applications, 8 (2010) 151-176, DOI
10.1007/s11784-010-0029-5,

**Abstract. **Period-doubling cascades are among the most prominent
features of many smooth one-parameter families of maps, F: R×M ®M ,
where M is a locally compact manifold without
boundary, typically R^{N}. In particular, we investigate F(µ,**·**) for µ in J
= [µ_{1}, µ_{2}], when F(µ_{1},** ·**) has only finitely many periodic orbits while F(µ_{2},**·**) has exponential growth of the number
of periodic orbits as a function of the period.
For generic F, under additional hypotheses, we use a fixed point index
argument to show that there are infinitely many ``regular'' periodic orbits at
µ_{2} (see Section 2) Furthermore, all but finitely many of these
regular orbits at µ_{2} are tethered to their own period-doubling
cascade. Specifically, each orbit ρ at µ_{2} lies in a connected
component C(ρ) of regular orbits in J×M, different regular orbits typically
are contained in different components, and each component contains a
period-doubling cascade. These components are one-manifolds of orbits, meaning
that we can reasonably say that an orbit ρ is ``tethered'' or ``tied'' to
a unique cascade. When F(µ_{2}) has horseshoe
dynamics, we show how to count the number of
regular orbits of each period, and hence the number of cascades in J×M. As corollaries of our main results,
we give several examples, proving that each has infinitely many cascades, and
counting the cascades of each period.

2011

**1. Argentine Ant** paper

A Draft Genome of the Globally Widespread and Invasive
Argentine ant (Linepithema humile),

Christopher Smith, followed by 48 names in alphabetical order incl.

James Yorke, Aleksey Zimin, and then Neil Tsutsui (UC
Berkeley)

Proceedings of the National Academy (PNAS) advance pub,

doi:10.1073/pnas.1007901108, January 31, 2011

Volume: **108**
Pages: **5673-5678** Published: **APR 5 2011**

**2**. Evelyn Sander and James A. Yorke,

Period-doubling cascades galore,

Ergodic Theory and Dynamics Systems,
31 (2011), 1249-1267

**3. **Steven L. Salzberg^{1,*},
Adam M. Phillippy^{2}, Aleksey Zimin^{3}, Daniela Puiu^{1},
Tanja Magoc^{1}, Sergey Koren^{2,4},
Todd Treangen^{1}, Michael C. Schatz^{5}, Arthur L. Delcher^{6},
Michael Roberts^{3}, Guillaume Marçais^{3}, Mihai Pop^{4},
and James A. Yorke^{3}

GAGE: a critical evaluation of genome assemblies and assembly algorithms,
Genome Research. http://genome.cshlp.org/conten

**
Abstract**. New sequencing technology has
dramatically altered the landscape of whole-genome sequencing, allowing
scientists to initiate numerous projects to decode the genomes of previously
unsequenced organisms. The lowest-cost technology can generate deep
coverage of most species, including mammals, in just a few days. The
sequence data generated by one of these projects consists of millions or
billions of short DNA sequences (reads) that range from 50-150 nucleotides in
length. These sequences must then be assembled

**C. Original Contributions in Symposium Proceedings and other Volumes **

1,2. J. A. Yorke, Spaces of solutions, and Invariance of contingent equations, both in Mathematical Systems Theory and Economics II, Springer-Verlag Lecture Notes in Operations Res. and Math. Econ. #12, 383-403 and 379-381: The Proceedings of International Conference for Mathematical Systems Theory and Economics in Varenna, Italy, June 1967.

3. J. A. Yorke, An extension of Chetaev's instability theorem using invariant sets, ibid. 100-106.

4. A. Halanay and J. A. Yorke, Some new results and problems in the theory of differential-delay equations, SIAM Rev. 13 (1971), 55-80: A revision of an invited report prepared for a conference in Czernowitz U.S.S.R., September 1968.

5. Selected topics in differential-delay equations, Japanese-United States Seminars on Ordinary Differential and Functional Equations, Springer-Verlag Lecture Notes in Math. #243, 1972, 17-38: The proceedings of a conference in Kyoto, September 1971.

6. S. A. Woodin and J. A. Yorke, Disturbance, fluctuating rates of resource recruitment, and increased diversity, in Ecosystem Analysis and Prediction, S. Levin, ed.: The proceedings of a SIMS conference held in Alta, Utah, July 1974, 1976, 38-41.

7. J. L. Kaplan and J. A. Yorke, Toward a unification of ordinary differential equations with nonlinear semi-group theory, International Conference on Ordinary Differential Equations, H. Antosiewicz, ed., Academic Press (1975), 424-433: The proceedings of a conference in Los Angeles, September 1974.

8. J. Curry and J. A. Yorke, A transition from Hopf
bifurcation to chaos: Computer experiments with maps in R^{2}, in The
Structure of Attractors in Dynamical Systems, Springer Lecture Notes in Math
#668, 48-66: The proceedings of the NSF regional conference in

An interesting more detailed paper is D. G. Aronson, M. A. Chory, G. R. Hall1 and R. P. McGehee, Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Commun. Math. Phys. 83 303-354 (1982).

9. J. Alexander and J. A. Yorke,
Parameterized functions, bifurcation, and vector fields on spheres, in Problems
of the Asymptotic Theory of Nonlinear Oscillations Order of the Red Banner,

10. J. L. Kaplan and J. A. Yorke, Numerical solution of a generalized eigenvalue problem for even mappings, in Functional Differential Equations and Approximation of Fixed Points, H. O. Peitgen and H. O. Walther, eds., Springer Lecture Notes in Math # 730 (1979), 228-237.

11. J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximation of Fixed Points, H. O. Peitgen and H. O. Walther, eds., Springer Lecture Notes in Math # 730 (1979), 204-227.

12. T. Y. Li and J. A. Yorke, Path following approaches for solving nonlinear equations: Homotopy, continuous Newton, projection, ibid, 257-264.

13. S. N. Chow, J. Mallet-Paret and J. A. Yorke, A homotopy method for locating all zeroes of a system of polynomials, ibid, 77-78.

14. E. D. Yorke and J. A. Yorke, Chaotic behavior and fluid dynamics, in Hydrodynamic Instabilities and the Transition to Turbulence, H. L. Swinney and J. P. Gollub, eds., Topics in Applied Physics 45 Springer-Verlag (1981), 77-95.

15. T. Y. Li and J. A. Yorke, A
simple reliable numerical algorithm for following homotopy paths, in Analysis
and Computation of Fixed Points, Academic Press (1980), 73-91: The proceedings
of Math.

16. J. C. Alexander, T. Y. Li and J. A. Yorke, Piecewise smooth homotopies, in Homotopy Global Convergence: The proceedings of the NATO Advanced Research Institute on Homotopy Methods and Global Convergence in Sardinia, June 1981, Plenum Publishing Corp. 1983, 1-14.

17. S. N. Chow, J. Mallet-Paret and J. A. Yorke, A bifurcation invariant: Degenerate orbits treated as clusters of simple orbits, in Geometric Dynamics, Springer Lecture Notes in Mathematics #1007 (1983), 109-131: The proceedings of a dynamics meeting at IMPA in Rio de Janeiro, August 1981.

18. J. Harrison and J. A. Yorke, Flows on S3 and R3 without periodic orbits, ibid, 401-407.

19. K. T. Alligood, J. Mallet-Paret and J. A. Yorke, An index for the global continuation of relatively isolated sets of periodic orbits, ibid, 1-21.

20. T. Short and J. A. Yorke, Truncated development of chaotic attractors in a map when the Jacobian is not small, in Chaos and Statistical Methods, Y. Kuramoto, ed., Springer-Verlag (1984), 23-30: The proceedings of the 6th Kyoto Summer Institute in September 1983.

21. C. Grebogi, E. Ott and J. A. Yorke, Quasiperiodicity and chaos, in Group Theoretical Methods in Physics, ed. W. W. Zachary (World Scientific, Singapore, 1984), pp. 108-110.

22. C. Grebogi, E. Ott and J. A. Yorke, N-Frequency quasiperiodicity and chaos in dissipative dynamical systems, in Proc. U.S.-Japan Workshop on Statistical Plasma Physics, Nagoya, Japan, 1984), pp. 71-74.

23. Wm. E. Caswell and J. A. Yorke, Invisible errors in dimension calculations: Geometric and systematic effects, in Dimension and Entropies in Chaotic Systems, ed., G. Mayer-Kress, Springer-Verlag Synergetic Series, 1986.

24. P. H. Carter, R. Cawley, A. L. Licht, M. S. Melnik and J. A. Yorke, Dimension measurements from cloud radiance, ibid.

25. C. Grebogi, E. Ott and J. A. Yorke, Fractal Basin Boundaries, Lecture Notes in Physics, Vol. 278 (The Physics of Phase Space), Springer-Verlag, (1986), 28-32.

26. C. Grebogi, E. Ott, H. E. Nusse and J. A. Yorke, Fractal basin boundaries with unique dimensions, in Chaotic Phenomena in Astrophysics, Vol. 497 of the Ann. New York Acad. Sci. (1987), 117-126.

27. C. Grebogi, H. E. Nusse, E. Ott and J. A. Yorke, Basic Sets: Sets that determine the dimension of basin boundaries, In Dynamical Systems, Proc. of Special Year at the University of Maryland, Lecture Notes in Mathematics, ed. J. Alexander, 1342, 220-250, Springer Verlag, Berlin, etc. (1988).

28. J. A. Yorke, Report of the Research Briefing Panel on Order, Chaos, and Patterns: Aspects of Nonlinearity, National Academy of Sciences (1987), (Committee report), 1-14.

29. C. Grebogi, E. Ott and J. A. Yorke, "Pointwise dimension and unstable periodic orbits", in Essays on Classical and Quantum Dynamics, H. Uberall, ed. (Gordon and Breach, 1991), 57-62.

30. E. Ott, C. Grebogi and J. A. Yorke, "Controlling chaotic dynamical systems, in CHAOS: Soviet-American Perspective on Nonlinear Science 1, Ed. D. K. Campbell (Am. Inst. of Physics, New York, 1990), 153-172.

31. Y-C. Lai, C. Grebogi and J. A. Yorke, "Sudden change in the size of chaotic attractors: How does it occur? in Applications of Chaos (1992), 441-456.

32. C. Grebogi, E. Ott, F. Varosi and J. A. Yorke, "Analyzing chaos, A visual essay in nonlinear dynamics", in Energy Sciences Supercomputing, U.S. DOE National Energy Research Computer Center (1990), 30-33.

33. J. A. Kennedy and J. A. Yorke, "The forced damped pendulum and the Wada property", in Continuum Theory and Dynamical Systems, Lecture Notes in Pure and Applied Mathematics, editor Thelma West, (Marcel Dekker, Inc.) (1993), 157-181.

34. L. Poon, S. Dawson, C. Grebogi, T. Sauer and J. A. Yorke, Shadowing in chaotic systems, in Dynamical Systems and Chaos (World Scientific) (1995).

35. Y.-C. Lai, C. Grebogi and J. A. Yorke, Intermingled basins and riddling bifurcation in chaotic dynamical systems, to appear in US-Chinese Conference on Recent Developments in Differential Equations and Applications (1997), 138-163.

36. J. Levine, P. Rouchon, G.-H.
Yuan, C. Grebogi, B. Hunt, E. Kostelich, E. Ott and J. A. Yorke, On the control
of US Navy cranes, Proceedings of the European Control Conf. (

37. G.-H. Yuan, B. R. Hunt, C. Grebogi, E. Kostelich, E. Ott and J. A. Yorke, Design and control of shipboard cranes, Proc. of the 16th ASME Biennial Conference on Mechanical Vibration and Noise, September 1997, Sacramento, CA.

38. J. A. Kennedy and J. A. Yorke, A Chaos Lemma with applications to Henon-like Difference Equations,

in Proceedings of the Fifth International Conference on Difference Equations, Temuco, Chile held Jan 2-7, 2000, Edited by S. Elaydi et al, Taylor and Francis, 2002, New York, pp.173-205.

39. M. S. Baptista, B. R. Hunt, C. Grebogi, E. Ott, J. A. Yorke, Control of shipboard cranes, Proc. of the IUTAM Symposium on Recent Developments in Nonlinear Oscillations of Mechanical Systems (March 1999, Hanoi, Vietnam), Nguyen Van Dao, E. J. Kreuzer (eds.), Kluwer Academic Publishers (2000).

40. D.J. Patil, I. Szunyogh, B.R. Hunt, E. Kalnay, E. Ott, and J.A.
Yorke, Using Large Member Ensembles To Isolate Local Low Dimensionality of
Atmospheric Dynamics, Proceedings on Symposium on Observations, Data
Assimilation, and Probabilistic Prediction, Amer. Met. Soc. Jan. 13-17, 2002
Orlando Fl.,

41. I. Szunyogh, A. V. Zimin, D.
J. Patil, B. R. Hunt, E. Kalnay, E. Ott, J. A. Yorke, On the dynamical basis
for targeting weather observations,

42. D.J. Patil, I. Szunyogh, A.V. Zimin, B.R. Hunt, E. Ott, E. Kalnay, and J.A. Yorke, Local Low Dimensionality and Relation to Effects of Targeted Weather Observations. Proceedings of the 7th Experimental Chaos Conference. Aug. 25-29, 2002; San Diego, CA, USA.

Editors: Visarath In, Ljupco Kocarev, Thomas L. Carroll, Bruce J. Gluckman, Stefano Boccaletti, and Jurgen Kurths. American Institute of Physics Proceedings Volume 676. Amer. Inst. for Physics, Melville, New York. ISBN: 0-7354-0145-4

D. Papers in Symposium Proceedings - Announcements of Papers in Section B

1. J. A. Yorke, Lyapunov functions and the existence of solutions tending to O, Seminar on Differential Equations and Dynamical Systems, edited by G. S. Jones, Springer Verlag Lecture Notes in Math. #60 (1968), 48-54.

2. J. A. Yorke, Asymptotic stability for functional differential equations, ibid, 65-75; and Extending Lyapunov's second method to non-Lipschitz Lyapunov functions, 31-36.

3. J. A. Yorke, Some extensions of Lyapunov's second method, Differential Integral Equations, J. Nohel, ed., SIAM, Philadelphia, 1969, 206-207.

4. J. A. Yorke, Non-Lipschitz Lyapunov functions, Proceedings of the Fifth International Conference on Non-Linear Oscillations, 2 (1971), 170-176: Kiev, USSR, held August 1969.

5. K. Cooke and J. A. Yorke, Equations modelling population growth, economic growth and gonorrhea epidemiology, in Ordinary Differential Equations, Academic Press, 1972, 35-53: The proceedings of a Naval Research Lab meeting in Washington, DC, June 1971.

6. T. Y. Li and J. A. Yorke, The "simplest" dynamical system, in Dynamical Systems, Vol. 2, Academic Press, 1976, 203-206, Cesari, Hale and LaSalle, eds.: The proceedings of an international symposium at Brown University, August 1974.

7. J. L. Kaplan and J. A. Yorke, Existence and stability of periodic solutions of x'(t) = f(x(t),x(t-1)), ibid 137-142.

8. R. B. Kellogg, T. Y. Li and J. A. Yorke, A method of continuation for calculating a Brouwer fixed point, in Fixed Points, S. Karamadian, ed., Academic Press, 1977, 133-147: The proceedings of a conference at Clemson University, June 1974.

9. A. Nold and J. A. Yorke, Modelling gonorrhea, in Dynamical Systems, Bednarek and Cesari, eds., Academic Press, 1977, 367-382: The proceedings of a conference in Gainesville, FL, March 1976.

10. J. L. Kaplan and J. A. Yorke, The onset of chaos in a fluid flow model of Lorenz, in Bifurcation Theory and Applications in Scientific Disciplines, Annals of N.Y. Academy of Sci. 316, 400-407: The proceedings of a New York Academy of Science meeting, New York City, November 1977.

11. N. Nathanson, G. Pianigiani,
J. Martin and J. A. Yorke, Requirements for perpetuation and eradication of
viruses in populations, in Persistent Viruses, Academic Press, 1978, 76-100:
The proceedings of

12. J. Mallet-Paret and J. A. Yorke, Two types of Hopf bifurcation points: Sources and sinks of families of periodic orbits, in Nonlinear dynamics, Annals of N.Y. Academy of Sci. 357, 300-304: The proceedings of a meeting in Manhattan in December 1979.

13. S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, Fractal basin boundaries in nonlinear dynamical systems, in Statistical Physics and Chaos in Fusion Plasmas 1, Ed. C. W. Horton and L. E. Reichl (Wiley, New York, 1984): The proceedings of the U.S.-Japan International Workshop on Chaotic Dynamics in Austin, Texas, November 1982.

14. J. C. Alexander and J. A. Yorke, Dimensions of attractors of chaotic systems, for the proceedings of an IEEE meeting in Baltimore, March 1983.

15. S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, An obstacle to predictability, in Proc. XIIIth Intl Colloq. on Group Theoretic Methods in Phys. (World Scientific Publ. Co., Singapore, 1984).

16. C. Grebogi, E. Ott and J. A. Yorke, Quasiperiodicity and chaos, In Proc. XIIIth Intern. Colloq. on Group Theoretic Methods in Phys. (World Scientific Publ. Co., Singapore, 1984).

17. C. Grebogi, E. Ott and J. A. Yorke, N-frequency quasiperiodicity and chaos in dissipative dynamical systems, in Proc. U.S.-Japan Workshop on Statistical Plasma Physics, Nagoya, Japan (1984).

18. E. Kostelich and J. A. Yorke, Lorenz cross sections and dimension of the double rotor attractor, in proceedings of the September 1985 dimension meeting in Pecos: Dimension and entropies in chaotic systems, ed, G. Mayer-Kress, Springer-Verlag Synergetic Series, 1986.

19. C. Grebogi, E. Ott and J. A. Yorke, Fractal basin boundaries, in the 1986 Springer-Verlag volume of the Proceedings of the Physics of Phase Space held in College Park in June 1986.

20. K. T. Alligood and J. A. Yorke, Fractal basin boundaries and chaotic attractors, Proceedings of Symposia in Applied Mathematics, Vol. 39 (1989), 41-55.

21. E. Kostelich and J. A.
Yorke, Using Dynamic Embedding Methods to Analyze Experimental data, in The
Connection Between Infinite Dimensional and Finite Dimensional Dynamical
Systems, ed. B. Nicolaenko.

22. K. T. Alligood and J. A. Yorke, Global implications of the implicit function theorem, in Chaos, Order and Patterns, eds. R. Artuso, P. Cvitanovic and G. Casati, Plenum Press, N.Y. (1991)

23. J. A. Yorke, Chaos and
scientific knowledge, proceedings of conference on "Individuality and
Cooperation Action" (ed. Joseph E. Earley),

24. T. Sauer and J. A. Yorke,
Shadowing trajectories of dynamical systems (with) In Computer Aided Proofs in Analysis
(Eds. K. R. Meyer and D. S. Schmidt), 229- 234. The

25. M. Ding, C. Grebogi and J. A. Yorke, Chaotic Dynamics, in The Impact of Chaos on Science and Society (1993), Ed. C. Grebogi and J. Yorke (United Nations University Press, Tokyo, 1997), 1-15.

26. S. Banerjee, G. H. Yuan, E. Ott and J. A. Yorke, Anomalous bifurcations in dc-dc converters: Borderline collisions in piecewise smooth maps, Power Electronic Specialists Conf., St. Louis, MO, IEEE (June 1997), pp. 1337-1344.

**E. Shards **

J. A. Yorke, Ph.D. Thesis Style, Letters to the Editor of the A.M.S. Notices, June 1986, 517.

J. A. Yorke, The Goal of Communicating, Letter to the Editor of the A.M.S. Notices January 1987, 44-5.

J. A. Yorke, Peer Review - Not as the Magna Carta Prescribed, Letters to the Editor of the A.M.S. Notices August 1987, 756-757.

J. A. Yorke, The Beauty of Order
and Chaos, exhibit at Fine Arts Museum of Long Island, co-curated by H. Bruce
Stewart and C. R. Cutietta-Olson, April 1 -

J. A. Yorke, A chaos art show,
"Radical Science Stuff", created by Glen Woodward of The Museum of
Discovery and Science,

J. A. Yorke, A chaos exhibition,
"A Chaos of Delight: Artists and Scientists Seek an Understanding of Their
World" at The Delaware Center for The Contemporary Arts, Feb. 2 -

J. A. Yorke, An exhibition on Capitol Hill in Washington D.C. on March 19, 1996, sponsored by the Coalition for National Science Funding to demonstrate examples (Controlling Chaos) of NSF funding and the need for continuance.

Z. You and J. A. Yorke, Book Review, Mathematical Go Chilling Gets the Last Point, by E. Berlekamp and D. Wolfe, A. K. Peters, MA, 1994, for SIAM Review (1996), Vol. 38, #3, 527-546.

J. A. Yorke and M. Hartl,
Commentary on Efficient Methods for Covering Material and Keys to Infinity, for
Notices of the

J. A. Yorke, Chaos, New Scientist 27 Sept. 2005, p 37.

J. Yorke, Response to Ruelle, Notices of the AMS, November 2009.

**Dissertations and Theses Directed **

Steven E. Grossman, Ph.D. in Mathematics, 1969

Dissertation: Stability and Asymptotic Behavior of Differential Equations.

Shui-Nee Chow, Ph.D. in Mathematics, 1970

Dissertation: Almost periodic Differential Equation

James Kaplan, Ph.D. in Mathematics, 1970

Dissertation: Some Results in Stability Theory for Ordinary Differential Equations

Stephen H. Saperstone, Ph.D. in Mathematics, 1971

Dissertation: Controllability of Linear Oscillatory Systems Using Positive Controls

Thomas Martin Costello, Ph.D. in Mathematics, 1971

Dissertation: Fundamental Theory of Differential and Integral Equations

Dissertation: On the Controllability and Observability of finite Dimensional Systems

Gina Bari Kolata, M.S. in Mathematics, 1972

Dissertation: A Mathematical Model of Chemical Relaxation to a Cooperative Biochemical Process

Ana Lajmanovich Gergely, Ph.D. in Mathematics, 1974

Dissertation: Mathematical Models and the Control of Infectious Diseases

Tien-Yien Li, Ph.D. in Mathematics, 1974

Dissertation: Dynamics for x_{n+1}
= F(x_{n})

Glenn Kelly, M.A. in Mathematics, 1974

Dissertation: The Kurzweil-Henstock Integral

Annett Nold, Ph.D. in Mathematics, 1977

Dissertation: Systems Approaching Equilibria in Disease Transmission and Competition for Resources

Ira Schwartz, Ph.D. in Mathematics, 1980

Dissertation: Proving the Existence of Unstable Periodic Orbits Using Computer-Based Estimates

Stephen Pelikan, Ph.D. in
Mathematics from

Dissertation: The Dimension of Attractors in Surfaces

Brian Hunt, M.A. in Mathematics, 1983

Dissertation: When All Solutions of x' = ... Oscillate

Tobin Short, M.S. in Applied Mathematics, January 1984

Dissertation: The Development of Chaotic Attractors in the Early Stages of Horseshoe Development

Frank Varosi, M.S. in Applied Mathematics, December 1985

Dissertation: Efficient Use of Disk Storage for Computing Fractal Dimensions

Eric Kostelich, Ph.D. in Applied Mathematics, December 1985

Dissertation: Basin Boundary Structure and Lorenz Cross Sections of the Attractors of the Double Rotor Map

Laura Tedeschini, Ph.D. in Applied Mathematics, June 1986

Dissertation: How Often Do Simple Dynamical Processes Have Infinitely Many Coexisting Sinks?

Peter Battelino, Ph.D. in Physics (co-advisors: Ott, Grebogi), 1987

Dissertation: Three-Frequency Periodicity, Torus Break-up, and Multiple Coexisting Attractors in a Higher Dimensional Dissipative Dynamical System

Zhi-Ping You, Ph.D. in Mathematics, l991

Dissertation: Numerical Study of Stable and Unstable Manifolds of Some Dynamical Systems.

Ying-Cheng Lai, Ph.D. in Physics (co-advisors: Ott, Grebogi), l992

Dissertation: Nonhyperbolicity in Classical and Quantum Chaos.

Troy Shinbrot, Ph.D. in Physics (co-advisors: Hunt, Ott), l992

Dissertation: Controlling Chaos: Using the Butterfly Effect to Direct Trajectories to Targets in Chaotic Systems.

Ivonne Diaz-Rivera, M. A. in Applied Mathematics, 1995

Scholarly Paper: Strange Attractor Reconstruction from Experimental Data: A Review

Wai Chin, Ph.D. in Math (co-advisors: Hunt, Ott), 1995

Dissertation: Chaotic Dynamics in Piecewise Smooth Systems.

In 2006 Associate Director for Biostatistics at Genzyme

Barry Peratt, Ph.D. in Math at

Dissertation: Mixing Powders and Scrambling Points.

Jacob Miller, Ph.D. in Math at

Dissertation: Finding Periodic Orbits of Maps: Basins of Attraction of Numerical Techniques.

Leon Poon, Ph.D. in Physics (co-advisors: Ott, Grebogi), 1996

Dissertation:
Shadowability, Complexity, and

Ali Fouladi, Ph.D. in Physics (co-advisor: Ott), 1996

Dissertation: Spatio-temperal Patterns and Chaos Control

Ernest Barreto, Ph.D. in Physics (co-advisor: Ott), 1996

Dissertation: Stability in Chaotic Systems

Guo-Hui Yuan, Ph.D. in Physics (co-advisors: Ott, Hunt), 1997

Dissertation: Shipboard Crane Control, Simulated Data Generation and Border - Collision Bifurcations

Guocheng Yuan, Ph.D. in Mathematics (co-advisors: Ott, Hunt), 1999

Dissertation: Properties of Numerical Experiments in Chaotic Dynamical Systems

Carl Robert, Ph.D. in Physics (co-advisors: Ott, Grebogi), 1999

Dissertation: Explosions in Chaotic Dynamical Systems: How New Recurrent Sets Suddenly Appear and a Study of their Periodicities

Josh Tempkin, Ph.D. in Mathematics, 1999

Dissertation: Spurious Lyapunov Exponents Computed Using the Eckmann – Ruelle Procedure

Mitrajit Dutta, Ph.D. in Physics (co-advisor: Ott), 2000

Dissertation: Chaotic Systems Predictable Unpredictabilities and Synchronization

David Sweet, Ph.D. in Physics (co-advisor: Ott), 2000

Dissertation: Higher Dimensional Non Linear Dynamical Systems: Bursting and Scattering

Dhanurjay (DJ) A.S. Patil, Ph.D. in Applied Mathematics, (co-advisors: Ott, Hunt, Kalnay) 2001

Dissertation: Applications of Chaotic Dynamics to Weather Forecasting

Linda J. Moniz, Ph.D. in Mathematics, 2001

Dissertation: Convergence of Dynamically Defined Upper Bounds Sets

Aleksey Zimin,
Ph.D. in Physics, 2003 (co-advisor;

Dissertation: The Bubbling Transition and Data Assimilation

Michael Roberts, Ph.D. in Computer Science, 2003 (Samir Khuller was the official CS adviser)

Dissertation: A Preprocessor for Shotgun Assembly of Large Genomes

Michael
Oczkowski, Ph.D. in Physics, 2003 (co-advisor;

Dissertation: Scenarios for the Development of Locally Low Dimensional Atmospheric Dynamics

Kathleen A. Meloney, M.A. in Mathematics, 2004

Thesis: A Dynamical Systems approach to Estimating the Sequences of Repeat Regions in the Genome

William Ott, Ph.D. in Mathematics, 2004 (co-advisors: B. Hunt & D. Levermore)

Dissertation: Infinite-Dimensional Dynamical Systems and Projections

Brandy L. Rapatski, Ph.D. in Applied Mathematics and Scientific Computation, 2004 (co-advisor: F. Suppe)

Dissertation: The Non-Linear Transmission Dynamics of HIV/AIDS

Cevat Ustun, Ph.D. in Physics, 2005 (co-advisor: B. Hunt)

Dissertation: Improving Genome Assembly

Ian Frommer, Ph.D. in Applied Math and Sci. Computing, 2005 (co-advisors: Hunt & Bruce Golden)

Dissertation: Modeling and Optimization of Transmission Networks

Ryan Lance, Ph.D. in Mathematics, 2005 (co-advisor: B. Hunt)

Dissertation:

Joseph Skufca, Ph.D. in Mathematics, 2005

Dissertation: Understanding a Chaotic Saddle with Focus On A 9-Variable Model of Planar Couette Flow

Christopher M. Danforth, Ph.D. in Applied Mathematics, 2006 (co-advisor: E. Kalnay)

Dissertation: Making Forecasts for Chaotic Processes in the Presence of Model Error

Suzanne S. Sindi, Ph.D. in Applied Mathematics, 2006 (co-advisor: B. Hunt)

Dissertation: Describing and
Modeling Repeated Sequences in

Amy Finkbiner, Ph.D. in Mathematics 2007 (co-advisors B. Hunt and D. Margetis)

Dissertation: Global phenomena from local rules: peer-to-peer networks and crystal steps

Poorani Subramanian, Ph.D. in Applied
Mathematics, Statistics, and Scientific Computation Program (AMSC) 2010 (co-advisor
Aleksey Zimin)

Dissertation: Detecting and Correcting Errors in Genome Assemblies

Guillaume Marcais, Ph.D. in AMSC, (co-advisor Carl Kingsford)

46 Ph.D.s

The nonlinear dynamics group
generally has students work with several faculty members and as a result
students can have multiple advisors.

**Postdocs
supervised (jointly with collaborators)**

Celso Grebogi

S. W. McDonald

Eric Kostelich

Brian Hunt

Silvina P. Dawson

Ernest Barreto

Myong-Hee Sung

Lyman Hurd

D.J. Patil

Crystal Cooper

Wayne Hayes, 11/2002 - 6/2004

Aleksey Zimin

Michael Roberts

Nandi Leslie

**Invited Lectures** (1975-present)

(Usually 1 hour unless otherwise stipulated)

MARCH 1975

Computer Science &
Biomathematics Meeting,

APRIL 1975

JUNE 1975

SEPTEMBER 1975

NOVEMBER 1975

JANUARY 1976

FEBRUARY 1976

APRIL 1976

JUNE 1976

JULY 1976

SEPTEMBER 1976

OCTOBER 1976

MARCH 1977

APRIL 1977

Northwestern University - Mathematics Department (2 Lectures)

JUNE 1977

Gordon Conference on Theoretical Biology

OCTOBER 1977

JANUARY 1978

JANUARY - FEBRUARY 1978

National Bureau of Standards (3 Lectures)

MARCH 1978

APRIL 1978

National Institutes of Health - Theoretical Biology

JULY 1978

NASA -

OCTOBER 1978

NOVEMBER 1978

Massachusetts Institute of Technology - Meteorology Department

JANUARY 1979

Georgia Tech - Mathematics Department

JUNE 1979

Gordon Conference on Theoretical Biology

Northwestern University - Global Dynamics Conference

AUGUST 1979

SEPTEMBER 1979

NOVEMBER 1979

The Johns

JANUARY 1980

Georgia Tech - Mathematics Department

JUNE 1980

OCTOBER 1980

Scripps & U.C.S.D. Nonlinear Feedback Conference

JANUARY 1981

JUNE 1981

NATO Meeting on Homotopies in

SEPTEMBER 1981

Naval

DECEMBER 1981

Courant Institute,

APRIL 1982

National Bureau of Standards

JUNE 1982

U.N.H./A.M.S. Ergodic Theory Meeting

FEBRUARY 1983

MARCH 1983

An organizer of

National Science Foundation - Mathematics Seminar

APRIL 1983

National Cancer Institute - Laboratory Theoretical Biology

University of Chicago - Mathematics and Physics

JUNE 1983

Haverford/NATO Experimental Chaos Meeting, Aberdeen Proving Ground

AUGUST 1983

A.M.S. Meeting, Albany - Biomathematics Program

SEPTEMBER 1983

NASA (Goddard) Colloquium

6th Kyoto, Japan Summer Institute: Statistical Physics and Chaos

OCTOBER 1983

Johns Hopkins University - Physics Colloquium

NOVEMBER 1983

National Bureau of Standards - Meeting on Fractals

JANUARY 1984

Dynamics Days meeting in San Diego

FEBRUARY 1984

Brown University - Applied Mathematics Seminar

MARCH 1984

Wisconsin University/Midwest Dynamical Systems Meeting -

American Physical Society (35 min. lecture)

APRIL 1984

Stevens Institute Physics Seminar

Naval Surface Weapons Center (8 Lectures)

JUNE 1984

OCTOBER 1984

Boston University - Science of Chaos Meeting

NOVEMBER 1984

Princeton Institute for Advanced Studies Meeting in Dynamics

MARCH 1985

Principal Lecturer, Georgia Tech Meeting (8 Lectures)

City College of City University - Physics Colloquium

APRIL 1985

Johns Hopkins University Physics Department (5 lectures)

City University of New York - Graduate School

Midwest Dynamical Systems

American Association of Physics Teachers, Annapolis

JUNE 1985

University of California - Nonlinear Sciences Meeting, UCLA

NOVEMBER 1985

Massachusetts Institute of Technology, Mathematics Department

Lehigh University, Department of Physics

JANUARY 1986

Dynamics Days Workshop, San Diego, CA

MARCH 1986

National Academy of Sciences, Washington, DC - International Workshop on AIDS

Princeton University, Applied Mathematics Colloquium

Harvard University - Condensed Matter Seminar

UMBC, Mathematics Department Colloquium

APRIL 1986

U.S. Naval Academy, Mathematics Department

Courant Institute, New York University - Seminar

NIH, Bethesda, MD - Conference on Perspectives in Biological Dyn. and Theor. Medicine - Meeting of Chaos in Biology

City

INRIA Workshop on Chaos and Turbulence, Paris, France - April 21-24 (10 hours of Lecture)

University of Tubingen, Germany - Institute for Information Sciences

California Institute of Technology

Philadelphia AAAS 1986 Annual Meeting, (1/2 hour lecture)

SEPTEMBER 1986

Cornell University, Mathematical Sciences Institute - Workshop on Nonlinear Dynamics

Pennsylvania State U., Dept. of Math. Dynamical Systems Conference

OCTOBER 1986

NASA Goddard Space Flight Center Lecture Series on "Advances in Computational Physics"

JANUARY 1987

La Jolla Institute, CA, Dynamic Days Conference

Monterey, CA, International Conference on Chaos in Physics

MARCH 1987

U.S. Naval Academy, Conf. on Modelling Advanced Technologies

APRIL 1987

Columbia U., New York City - Second Symposium on Complexity of Approximately Solved Problems

University of Houston, Physics Department (2 Lectures)

University of Texas at Austin - Physics Seminar

JUNE 1987

University of Missouri-Columbia, Conference on Computer Experimentation in Nonlinear Analysis

Syracuse, New York - DARPA Workshop on Parallel Architecture

JULY 1987

Joint

SEPTEMBER 1987

U of Cincinnati, OH - CBMS Regional Conference on Fractal Geometry

OCTOBER 1987

Yale University - Applied Mathematics Colloquium

Naval Surface Warfare Center Seminar

DECEMBER 1987

Massachusetts Institute of Technology - Lorenz Symposium

Naval Surface Warfare Center, Seminar

JANUARY 1988

Houston, Texas - Dynamics Days

FEBRUARY 1988

MARCH 1988

Columbus, OH - International Conference on Differential Equations

APRIL 1988

Syracuse, NY - Annual Joint Physics-Mathematics Colloquium

National Bureau of Standards, Gaithersburg, MD - Minicourse on Chaos

Northwestern Univ., Evanston, IL - Midwest Dynamical Systems Conference

Mitre Corp., McLean, VA

JUNE 1988

Utrecht, The Netherlands - Symposium on Chaotic Dynamical Systems (4 lectures)

Dusseldorf, West Germany - Dynamics Days

JULY 1988

Minneapolis, MN, presenter of SIAM Short Course (with J. Guckenheimer)

AUGUST 1988

SEPTEMBER 1988

Naval Surface Warfare Center, White Oak, MD

OCTOBER 1988

Johns Hopkins University, Applied Physics Lab., Laurel, MD - Keynote Speaker

Princeton Plasma Physics Lab., Princeton, NJ - Colloquium

Catholic University, Washington, DC - Saenz Symposium

University of Indiana, Bloomington - Mathematics Colloquium

NOVEMBER 1988

Vanderbilt University, Nashville, TN - Shanks Lecture Series (2 lectures)

JANUARY 1989

Plasma Physics Lab, Princeton, NJ, Conference on Graphics

MARCH 1989

Georgia Institute of Technology, Atlanta, GA

Auburn University, Auburn, AL - Southeastern Spring Dynamical Systems Conference

University of Cincinnati, Cincinnati, OH - Conference on Computer Aided Proofs in Analysis

APRIL 1989

Georgetown University - Conf. on Individuality & Cooperative Action

University of Maryland - Conf. on Physics for Students

JUNE 1989

University of Rhode Island, Kingston - Mathematics Department Colloquium

University of California, San Diego - 3rd Joint ASCE/ASME Mechanics Conference

Ames Research Center, Moffet Field, CA - Workshop on "Chaotic Dynamics" (4 Lectures)

OCTOBER 1989

George Washington University - Lecture in Philosophy Department

Smithsonian Institution - Conference on "Patterns in Chaotic Systems"

University of Maryland - CHPS Lecture

NOVEMBER 1989

Eleanor Roosevelt High School, Greenbelt, MD

DECEMBER 1989

Rutgers University, Mathematics Department - Seminar

University of Maryland at College Park - Physics Colloquium

MARCH 1990

Towson State University - Physics Colloquium

Montgomery Blair High School, Takoma Park - Montgomery County Physics Teachers

University of Michigan - Symposium on Applications of Nonlinear Studies

APRIL 1990

University of Maryland at College Park - Conference on Low Dimensional Dynamics

Carleton University, Ottawa, Canada - Pure and Applied Analysis Day

Northwestern University - Mathematics Colloquium

National Institutes of Health,

Orlando - SIAM Conference on Applications of Dynamical Systems

Los Alamos - Conference on Nonlinear Science: "The Next Decade"

George Washington University Hospital Lecture: "Chaos"

JUNE 1990

JULY 1990

Chicago- SIAM Annual Meeting - presented Short Course

AUGUST 1990

University of Maryland, College Park - American Association of Physics

Teachers, US/USSR Physics Student Exchange Program

SEPTEMBER 1990

Princeton University, Princeton, NJ - Mathematics Department - Colloquium

Bryn Mawr College - Math Department Colloquium

OCTOBER 1990

Rockville, Maryland, Strathmore Hall Arts Center - "Voices of our Time Series"

Midwest Dynamical Systems Seminar at University of Cincinnati

NOVEMBER 1990

Philadelphia - 12th Annual International Conference, IEEE Symposium on Chaos and Fractals

UMCP, College of Behavioral & Social Science

DECEMBER 1990

University of Delaware - Mathematics Department Colloquium

JANUARY 1991

Houston, Texas - Dynamics Days Texas

National Institutes of Health, Bethesda, Maryland

FEBRUARY 1991

University of MD, College Park - Undergraduate Mathematics Colloquium

MARCH 1991

Naval Research Laboratory, Washington, DC

University of Maryland, College Park - Department of Textiles and Consumer Economics

APRIL 1991

U of MD - 8th International Conference on Mathematical and Computer Modelling - Keynote Speaker

The

Triangle Park, NC - Army Research Office Workshop on Fractals and Chaos

JUNE 1991

Berlin, Germany - Dynamic Days

AUGUST 1991

OCTOBER 1991

Pennsylvania State University - Dynamical Systems and Related Topics

Montana State University - Midwest Dynamical Systems Conference

JANUARY 1992

Oberwolfach, Germany, Conference on Applied Dynamics and Bifurcation

MARCH 1992

Courant Institute, Mathematics Colloquium

UMCP, Maryland Junior Science and Humanities Symposium

Yale University, Mathematics Department

APRIL 1992

University of South Alabama, Mobile, Public Lecture

NSWC, Silver Spring, MD, Dynamics Day and a Half mini-symposium,

Carleton University, Ottawa, Canada, 14th Annual Analysis Day

UMCP Dance Department Colloquium

Rutgers University, Joint Physics/Dance Department Colloquium

Georgia Institute of Technology, Colloquium

JUNE 1992

Woudschoten,
The

National
Security Agency,

JULY 1992

Boston University, Regional Institute in Dynamical Systems

AUGUST 1992

Orlando, Florida - World Congress of Nonlinear Analysts, 2 lectures

SEPTEMBER 1992

Minneapolis, Minnesota, SIAM Conference on Control and its Applications

OCTOBER 1992

Snowbird,

NOVEMBER1992

University of Kentucky, Lexington, Midwest Southeastern-Atlantic Second Joint Conference on Differential Equations Conference, Keynote address

DECEMBER 1992

UMBC, National Mathematics Honor Society, Pi Mu Epsilon, Induction Speaker

Rockville, MD., C.E. Smith Jewish Day School, NEH Seminar on "Order and Chaos"

JANUARY 1993

Arizona State University at Tempe, "Dynamics Days" Present Lecture and a Short Course (6 Lectures), with C. Grebogi

FEBRUARY 1993

Albuquerque, NM, Sandia

MARCH 1993

University of South Carolina, Columbia, Spring Topology Conference

Goucher College, MD, invited lecture

APRIL 1993

George Mason University

Princeton University, N.J., Department of Mathematics, (talk, joint with Brian Hunt)

UMCP, Dept. of Physics, Graduate Students Seminar

SUNY, Stonybrook, "Dynamical System Seminar"

JULY 1993

San Diego, CA, SPIE, Keynote Speaker and Short Course (6 lectures)

AUGUST 1993

AT&T, New Jersey Chaos Seminar

UMCP, Phi Beta Kappa Consortium with the D.C. Schools

Budapest, International Conf. on Complex Geometry in Nature

SEPTEMBER 1993

Como, Italy, NATO Conference on Chaos Order & Patterns: Aspects of nonlinearity

OCTOBER 1993

Arlington, VA, 2nd Experimental Chaos Conference

Penn State Univ., "Semi-annual Regional Workshop in Dynamical Systems"

Howard Univ., Dynamical Systems Week

NOVEMBER 1993

NIH,
"Dynamical Systems Methods for the Study of Interactions of Genes and
Environment"

MARCH 1994

Riverside, CA, Univ. of Calif., Conference Statistical Mechanics

APRIL 1994

Stony Brook, N.Y., SUNY, North East Dynamical System Conference

Tokyo
Metropolitan University, Japan - Int'l Conf. on Dynamical Systems and Chaos (2
talks)

JUNE 1994

Amsterdam, Vrije University (2 talks)

Amsterdam, Dynamical Systems Lab, CWI

Amsterdam, "Chaotic Dynamical Systems Conference" (Woudschoten) (4 talks)

Budapest, "Dynamical Days - Budapest"

Johns Hopkins University, International Conference Chaos Theory in Psychology and the Life Sciences

Montgomery College, Takoma Park, NSF Teachers Institute.

NOVEMBER 1994

Georgia Inst. of Tech., Army Res. Workshop, "Nonlinear Dynamics in Sci"

JANUARY 1995

FEBRUARY 1995

Hurst,
TX,

Denton, TX, University of North Texas

MARCH 1995

Univ. of Del., 1995 Joint Spring Topology & Southeast Dynamics Conference

UMBC, National Math Honor Society (Induction Lecture)

Snowbird, Utah, SIAM Conf. on Dynamical Systems (Short Course and Talk)

JUNE 1995

Bryn Mawr College, Int'l Workshop: Measure of Spatio-Temporal Dynamics

Seattle, WA, Univ. of Washington, Workshop in Dimensional Theory and Dynamical Systems

AUGUST 1995

Old Town Alexandria, VA, Symmetry Conference: Natural & Artificial (45 min)

OCTOBER 1995

Annapolis, MD, US Naval Academy, Graduate Mathematics Colloquium

NOVEMBER 1995

Atlanta, GA, Center for Disease Control, Div. of Sexually Transmitted Disease Prevention

DECEMBER 1995

Georgia Tech. Conf. on Dynamical Numerical Analysis (40 minutes)

JANUARY 1996

Courant Institute, Workshop on Advances in Dynamical Chaos (40 min.)

FEBRUARY 1996

Towson State University, 1996 Spring Physics Seminar Series

MARCH 1996

Univ. of California, LA, Dynamical Systems Conference

APRIL 1996

Univ. of Wisconsin, Milwaukee (2 talks), Marden Lecture and Seminar

JUNE 1996

AUGUST 1996

Berlin, Germany, WE-Heraeus Seminar

OCTOBER 1996

Johns Hopkins University, (Physics Seminar)

University of Maryland, (Math Seminar)

Penn State University, Dynamical Systems Workshop

NOVEMBER 1996

Math Association of America, Hood College, Frederick, MD

Duke University, Durham, NC

National Academy of Sciences, Wash, D.C.

Stony
Brook Dynamical Seminar, Stony

DECEMBER 1996

Univ. of Delaware, Newark, DE

JANUARY 1997

Dynamics Days--Arizona, Scottsdale, AZ (1/2 hr. lecture)

FEBRUARY 1997

MURI Mtg., Virginia Tech, Blacksburg, VA (1/2 hr. lecture)

MARCH 1997

International Conference on Order and Chaos, Nihon Univ., Tokyo, Japan (50 minute lecture)

APRIL 1997

SIAM Meeting, Snowbird,

Drexel University, Philadelphia, Pennsylvania

JUNE 1997

NSF/CBMS
Conference,

Middle East Technical University, Ankora, Turkey (5 lectures)

NICHD Meeting on Fetal development, Rockville, MD (1/2 hour lecture)

JULY 1997

Conference on Nonlinear Phenomena in Dynamical Systems, Yukon, Canada

AUGUST 1997

MURI Meeting, Virginia Tech., Blacksburg, VA (1/2 hour lecture)

SEPTEMBER 1997

OCTOBER 1997

GA Tech, Topology Mtg., Atlanta, GA (contributed talk/20 minutes

Univ. of Wisconsin (Milwaukee)

NOVEMBER 1997

Howard Univ., Wash., D.C.

DECEMBER 1997

Weizmann Institute, Israel 2 lectures

Rehoboth 3 lectures

JANUARY 1998

Spring
Topology Mtg.,

Computational Science Initiative, OER, Gaithersburg, MD

FEBRUARY 1998

International Winter School, Max Planck Institute, Dresden, Germany

MARCH 1998

Penn
State/UMCP Dynamics Mtg.,

APRIL 1998

University of Texas at Austin, Physics Department

JULY 1998

Conférence on Dynamical Systems : Crystal
to Chaos, Maisel-Institut de Provence, Marseilles, France

Max Planck Institute, Dresden, Germany 2 Talks

University of Bayreuth, Physics Department, Germany

SEPTEMBER 1998

Conference on Computational Physics, Granada, Spain

OCTOBER 1998

MURI Meeting, Virginia Tech, Blacksburg, VA

Penn State University, Dynamical Systems Workshop

Los Alamos National Laboratory, Distinguished Lecture Series in Nonlinear Science, Los Alamos, NM

NOVEMBER 1998

Courant Institute, Nonlinear Dynamics Meeting, New York

George
Mason University, Krasnow Institute, Nonlinear Science Seminar,

JANUARY 1999

NSF/
Meeting on Mathematics and Molecular Biology VI,

MARCH 1999

MURI Meeting, Virginia Tech,

New Jersey Institute of Technology,

APRIL 1999

Los Alamos National Laboratory, Los Alamos, NM

SIAM Meeting, Snowbird,

JULY 1999

Dynamics Days Asia-Pacific, Hong Kong Baptist
University,

The Chinese University of Hong Kong,

The Advanced Crane Technology Workshop, Norfolk, VA (attendee?)

Pacific Institute for the Math.
Sciences, University of British Columbia,

Summer Institute on Smooth Ergodic
Theory and Applications, University of Washington,

AUGUST 1999

Bagozici University,

OCTOBER 1999

Brown
University,

Fall
Leadership Forum, Sandia National Laboratories,

Conference
on Chaos in Physics,

JANUARY 2000

Conference
on Differential Equations, Univ. of LaFrontera,

Joint
SIAM Meeting with

MARCH 2000

International Conference on Fundamental Sciences: Math/Theoretical Physics, Singapore (2 talks)

APRIL 2000

British Applied Math Colloquium
2000,

JUNE 2000

Dynamics Days, Surrey University,

AUGUST 2000

SIAM Pacific Rim Dynamical Systems
Conference, SIAM,

OCTOBER 2000

International
Workshop on Chaos & Nonlinear Dynamics, Kansai Univ.,

Dynamical
Systems Workshop,

JANUARY 2001

Dynamics Days 2001, Duke University, Chapel Hill (½ hr.)

MARCH 2001

Spring Topology and Dynamical
Systems Conference,

Recomb Satellite Meeting on

SIAM
Conf. on Applications of Dynamical Systems, Snowbird,

Symposium in Real Analysis XXV, Weber State Utah

AUGUST 2001

Conley Index at Sherbrooke Canada (½ hr.)

OCTOBER 2001