**CURRICULUM VITAE**

April 2004

**JAMES A.
YORKE**

Distinguished
University Professor of Mathematics and Physics

Institute for Physical Science and Technology

Web pages: www.chaos.umd.edu and www.ipst.umd.edu/~yorke/

e-mail: yorke@ipst.umd.edu

phone: 301-405-4875 fax: 301-314-9363

Born

**Education**

**Professional Positions**

Appointments
in the IPST (“IPST” denotes the Institute for Physical Science and Technology
or in its predecessors, at the

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

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

**Honors and Awards**

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 U of

U. of Md. Chaos Group rated #1
(as an area of physics in 1999 by U.S. News)

An APS Centennial Speaker - 1998-99

AAAS Fellow - elected 1998

First UMCP recipient of the University of Maryland Regents Faculty Award for
Excellence

in Research/Scholarship -
1998

38^{th} Annual Chaim Weizmann Memorial Lecturer

- Weizmann Institute Rehovot,
Israel - 1997

Distinguished University Professor - appointed 1995** **

Guggenheim fellow 1980

** **

**Editorial
boards**

International Journal of
Bifurcation and Chaos

Journal of Complex Systems

Chaos, Solitons and Fractals

Journal of Difference Equations and Applications

** **

**Principle
investigator on current research grants **

NSF
grants 2001-2006 Applications of Nonlinear Dynamics

NIH grant 2003-6 Reliable Assembler for Whole Genome Shotgun Data

LUCITE (and successors) Modeling Complex Data Networks
2002-present

**Membership**

Amer. Math Soc.

Amer. Phys. Soc.

Math. Assoc. of Amer.

**Current Research
Projects**

See: http://www.ipst.umd.edu/~yorke/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(x_{T}) 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.

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 R^{n} 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 {t_{i}, 0 < t_{i} < t_{i+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(t_{i})) 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 Aj 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 m_{k} converges to a smooth measure m_{0}
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 x_{0}
in I and x_{i+1} = f (x_{i}) for i 0. We say there is no
division for (x_{1}, x_{2},...,x_{n}) if there is no a
in I such that x_{j} < a for all j even and x_{j} < 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 T^{k} in R X T^{k}
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.

5.
B. R. Hunt and J. A. Yorke, When all solutions of dx/dt = Σ_{i} q_{i}(t)x(t-T_{i}(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 q_{i}(t) and T_{i}(t) are positive. In
particular, it is shown that if liminf sum_{i }= ln T_{i}(t)q_{i}(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 L^{1} 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 L^{1} with f = 0 and the norm of f = 1,
P^{n}f converges to f when n goes to infinity, where F is a strongly
compact subset of L^{1}. It follows from this decomposition that P^{n}f
is asymptotically periodic for any f in L^{1}.

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 A_{g} has
positive volume. We conjecture that the generalized volume of A_{g} is
proportional to g^{D-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 = k_{1} exp [k_{2}(a-a_{0})^{-1/
2}]

for
a near a_{0}, where k_{1} and k_{2} are constants and a_{0}
is the value of the parameter a at which the crisis occurs. Thus, as a
approaches a_{0} from above, T increases more rapidly than any power of
(a-a_{0})^{-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

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,

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 D_{q} 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 g_{m} for either f_{m}
or F_{m} with f_{m}(x) = mx for x <= 1/2 and f_{m}
(x) = m(1-x) for x >= 1/2, and F_{m}(x) = mx (1-x). For a given m we
will say: g_{m} 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 g_{m}-orbit starting in some invariant
interval can be deltax -shadowed by a real g_{a} -orbit with a = m +
delta m. We show that g_{m} typically permits increased parameter
shadowing.

4.
H. E. Nusse and J. A. Yorke, Period halving for x_{n+1 }= MF(x_{n})
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 STD clinic has reported a 20-fold drop in the anal/rectal gonorrhea incidence rate over a 6-year period (1981-1987). By using a gonorrhea epidemiology model we find that a 33% drop in the infectious contact rate is sufficient to explain the observed decrease in anal/rectal gonorrhea.

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, D_{q}
(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<q_{T}) 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-III intermittencies. We examine the scaling of
the dimension of basin boundaries near these intermittency transitions. We
find, from numerical experiments, that near the transition the dimension scales
with a system parameter p according to the power law D is asymptotically like d_{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-III intermittency d_{0} = D, where
D is the dimension of the space. Heuristic analytic arguments supporting the
above are presented.

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 ƒ_{λ}
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*) = λ*x*(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

* *

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 R^{k}. 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 E_{m} 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 E_{m}F 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,

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 10^{11} 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.

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, and 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) E_{m} 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 f_{m}
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 f_{a}(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,

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 R2, 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,

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 C^{4} 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 B_{1}, B_{2},...,
B_{N} (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 R^{N} 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 M_{n} 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 D_{1}
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-D_{1}) 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 epsilon^{1/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 F_{n}.

3.
D. Sweet, E. Ott and J.A. Yorke, Topology in chaotic scattering, Nature, 399 (

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, J.A. Yorke and G.-C. Yuan, Multiple attractor bifurcations: a source of unpredictability in piecewise smooth systems, Phys. Rev. Lett., 83 (1999), #21, 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

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}) 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 DNA sequences across the three domains of life, Physica D 146
(2000), pp. 388-396.

Abstract: We report statistical studies of correlation properties of ~7500 gene sequences, covering coding (exon) and non-coding (intron) sequences for DNA and primary amino acid sequences for proteins, across all three domains of life, namely Eukaryotes (cells with nuclei), Prokaryotes (bacteria) and Archaea (archae-bacteria). Mutual information function, power spectrum and Hölder exponent analyses show exons with somewhat greater correlation content than the introns studied. These results are further confirmed with hypothesis testing. While ~30% of the Eukaryote coding sequences show distinct correlations above noise threshold, this is true for only ~10% of the prokaryote and Archaea coding sequences. For protein sequences, we observe correlation lengths similar to that of “random” sequences.

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), #20 pp. 4265-4268.

** **

No abstract.

**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.

No Abstract.

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.

Abstract: A basic question
always present when obtaining numerical solutions is to what extent they are
valid. This question is especially
meaningful when dealing with chaotic dynamics, since local sensitivity to small
errors is the hallmark of a chaotic system.
Floating-point calculations commonly used to approximate solutions of
differential equations or compute discrete maps produce *pseudo-trajectories*,
which differ from true trajectories by new, small errors at each computational
step. Despite the sensitive dependence
on initial conditions, the methods of shadowing have shown that for chaotic
systems that are hyperbolic or nearly hyperbolic, locally sensitive
trajectories are often *globally insensitive*, in that there exist true
trajectories with adjusted initial conditions, called shadowing trajectories,
very close to long computer-generated pseudo-trajectories. A dynamical system is hyperbolic if phase
space can be spanned locally by a fixed number of independent stable and unstable
directions that are consistent under the operation of the dynamics.

In the absence of hyperbolic
structure, much less is known about the validity of long computer
simulations. Recently it was shown that
trajectories of a chaotic system with a fluctuating number of positive finite-time
Lyapunov exponents fail to have long shadowing trajectories. In other words, they are globally sensitive
to small errors. Such “hyperchaotic”
system has two positive Lyapunov exponents, although finite-time approximations
of the smaller of the two fluctuate about zero, due to visits of the trajectory
to regions of the attractor with a varying number of stable and unstable
directions. The destruction of
hyperbolicity caused by this phenomenon leads to global sensitivity – only
relatively short pseudo-trajectories will be approximately matched by true
system trajectories.

Our discussion of the global
sensitivity of trajectories for these non-hyperbolic systems is limited in this
review to the comparison between physical models and computer simulations, but
the same questions arise whenever comparing the time behavior of two systems
evolving under similar, but slightly different dynamical rules. For example, a natural system and its
theoretical *model* differ by modeling errors. In the presence of fluctuating Lyapunov exponents, global
sensitivity may lead to trajectory mismatch, in particular when long times are
considered. The result is that no
trajectory of the theoretical model matches, even approximately, the true system
outcome over long time spans.

**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-.

3.
K. Alligood, E.
Sander, and J. Yorke, Explosions: global bifurcations at heteroclinic
tangencies, Ergodic Theory and Dynamical Systems, Volume 22, Issue 4, Pages
953-972, 2002.

**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.

The paper has no abstract.

** **

**2003**

** **

**1**. William Ott and James
A. Yorke, Learning About Reality From Observation, SIAM Journal on Applied
Dynamical Systems, 2003, in press.

**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 : 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 R^{m}
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., 2003 **23** 895-906.

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, in press.

I. Szunyogh, A.V. Zimin, DJ Patil, BR Hunt, E.
Kalnay, E. Ott, and JA Yorke, On the Dynamical Basis of Targeting Weather
Observations, Proceedings on Symposium on Observations, Data Assimilation, and
Probabilistic Prediction, Amer. Met. Soc.

Michael Roberts, Brian R. Hunt, and James A. Yorke, Randall
Bolanos, and Art Delcher, A Preprocessor for Shotgun Assembly of Large Genomes,
J Comp Biology, in press.

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.

**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

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,

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

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

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 Fargo, ND, June 1977.

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

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

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 S^{3} and R^{3} 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.

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,

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,

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. (ECC 97), July 1997.

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

38.
Celso Grebogi, Leon Poon, Tim Sauer, J. A. Yorke, Ditza Auerbach, Shadowability
of Chaotic Dynamical Systems, Chapter 7 in Handbook of Dynamical Systems, Vol
2, 2002. Edited by B Fiedler. Elsevier Science B. V. (ISBN: 0-444-50168-1)

39. 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.

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.

41.
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.

Editors:
Visarath In, Ljupco Kocarev, Thomas L. Carroll, Bruce J. Gluckman, Stefano
Boccaletti, and Jurgen

Kurths. American Institute of Physics Proceedings Volume 676. American
Inst. for Physics,

**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,

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 ICN-UCLA Symposium on Persistent Viruses, Keystone, CO, March 1978. This paper is essentially a preliminary version of Journal Paper #59.

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

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 IMA Volumes in Mathematics and its Applications, Vol. 28,

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 **

Letters to the Editor of the A.M.S. Notices:

Ph.D. Thesis Style, June 1986, 517.

The Goal of Communicating, January 1987, 44-5.

Peer Review - Not as the Magna Carta Prescribed, 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 AMS (1997), Vol. 44, #66, 685-687.

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 ^{1,2}, 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 ^{1,2,} l992

Dissertation: Nonhyperbolicity in Classical and Quantum Chaos.

Troy
Shinbrot, Ph.D. in Physics ^{1,2}, 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 ^{1,3,} 1995

Dissertation: Chaotic Dynamics in Piecewise Smooth Systems.

Barry
Peratt, Ph.D. in Math at ^{4}, 1996

Dissertation: Mixing Powders and Scrambling Points.

Jacob
Miller, Ph.D. in Math at ^{4}, 1996

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

Leon
Poon, Ph.D. in Physics ^{1,2}, 1996

Dissertation:
Shadowability, Complexity, and

Ali
Fouladi, Ph.D. in Physics ^{2}, 1996

Dissertation: Spatio-temperal Patterns and Chaos Control

Ernest
Barreto, Ph.D. in Physics ^{2}, 1996

Dissertation: Stability in Chaotic Systems

Guo-Hui
Yuan, Ph.D. in Physics ^{2,3}, 1997

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

Guocheng
Yuan, Ph.D. in Mathematics ^{2,3,} 1999

Dissertation: Properties of Numerical Experiments in Chaotic Dynamical Systems

Carl
Robert, Ph.D. in Physics ^{1,2,} 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 ^{2}, 2000

Dissertation: Chaotic Systems Predictable Unpredictabilities and Synchronization

David
Sweet, Ph.D. in Physics ^{2}, 2000

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

Dhanurjay
(DJ) A.S. Patil, Ph.D. in Applied Mathematics^{2,3,5}, 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 with

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 with

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

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

1 supervised jointly with C. Grebogi

2 supervised jointly with E. Ott

3 supervised jointly with B. Hunt

4 supervised jointly with J. A. Kennedy

5 supervised jointly with E. Kalnay

**Postdocs
supervised (jointly with collaborators)**

Celso Grebogi

S. W. McDonald

Eric Kostelich

Brian Hunt

S. P. Dawson

Ernest Barreto

Myong-Hee Sung

Lyman Hurd

D.J. Patil

Crystal Cooper

Wayne Hayes

**Invited
Lectures** (1975-present)

(Usually 1 hour unless otherwise stipulated)

MARCH 1975

Computer Science & Biomathematics
Meeting,

APRIL 1975

MAY 1975

JUNE 1975

SEPTEMBER 1975

NOVEMBER 1975

JANUARY 1976

FEBRUARY 1976

APRIL 1976

JUNE 1976

JULY 1976

IBM

SEPTEMBER 1976

OCTOBER 1976

MARCH 1977

APRIL 1977

Northwestern University - Mathematics Department (2 Lectures)

MAY 1977

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

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

MAY 1982

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

JUNE 1983

Haverford/NATO Experimental Chaos
Meeting,

AUGUST 1983

A.M.S. Meeting,

SEPTEMBER 1983

NASA (Goddard) Colloquium

6th Kyoto Summer Institute: Statistical Physics and Chaos

OCTOBER 1983

NOVEMBER 1983

National Bureau of Standards - Meeting on Fractals

JANUARY 1984

Dynamics Days meeting in

FEBRUARY 1984

MARCH 1984

Wisconsin University/Midwest Dynamical Systems Meeting -

American Physical Society (35 min. lecture)

APRIL 1984

Stevens Institute Physics Seminar

MAY-JULY 1984

Naval Surface Weapons Center (8 Lectures)

JUNE 1984

OCTOBER 1984

NOVEMBER 1984

Princeton Institute for Advanced Studies Meeting in Dynamics

MARCH 1985

Principal Lecturer, Georgia Tech Meeting (8 Lectures)

APRIL 1985

Johns Hopkins University Physics Department (5 lectures)

MAY 1985

American Association of Physics
Teachers,

JUNE 1985

NOVEMBER 1985

Massachusetts Institute of Technology, Mathematics Department

1986

JANUARY 1986

Dynamics Days Workshop,

MARCH 1986

National

UMBC, Mathematics Department Colloquium

APRIL 1986

Courant Institute,

NIH,

City

INRIA Workshop on Chaos and
Turbulence,

MAY 1986

California Institute of Technology

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

SEPTEMBER 1986

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

OCTOBER 1986

JANUARY 1987

La Jolla Institute, CA, Dynamic Days Conference

MARCH 1987

APRIL 1987

MAY 1987

JUNE 1987

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

JULY 1987

Joint AMS-SIAM Summer Research Conference

SEPTEMBER 1987

U of

OCTOBER 1987

Naval

DECEMBER 1987

Massachusetts Institute of Technology - Lorenz Symposium

Naval

JANUARY 1988

FEBRUARY 1988

MARCH 1988

APRIL 1988

National Bureau of Standards,

Northwestern University,

MAY 1988

Mitre Corp.,

JUNE 1988

JULY 1988

AUGUST 1988

SEPTEMBER 1988

Naval

OCTOBER 1988

NOVEMBER 1988

JANUARY 1989

Plasma Physics Lab,

MARCH 1989

Georgia Institute of Technology,

University of

APRIL 1989

JUNE 1989

AMS Short Course "Chaos
'89" in

OCTOBER 1989

Smithsonian Institution - Conference on "Patterns in Chaotic Systems"

NOVEMBER 1989

DECEMBER 1989

MARCH 1990

APRIL 1990

Northwestern University - Mathematics Colloquium

MAY 1990

National Institutes of Health,

JUNE 1990

JULY 1990

AUGUST 1990

Teachers, US/USSR Physics Student Exchange Program

SEPTEMBER 1990

OCTOBER 1990

NOVEMBER 1990

UMCP, College of Behavioral & Social Science

DECEMBER 1990

JANUARY
1991

National
Institutes of Health,

FEBRUARY 1991

MARCH 1991

Naval
Research Laboratory,

APRIL 1991

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

MAY 1991

The

JUNE
1991

AUGUST
1991

OCTOBER
1991

JANUARY 1992

MARCH 1992

Courant Institute, Mathematics Colloquium

UMCP,

APRIL
1992

NSWC,

MAY 1992

UMCP
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

SEPTEMBER 1992

OCTOBER 1992

Snowbird,

NOVEMBER1992

on Differential Equations Conference, Keynote address

DECEMBER 1992

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

JANUARY
1993

FEBRUARY 1993

MARCH
1993

APRIL
1993

MAY 1993

UMCP, Dept. of Physics, Graduate Students Seminar

SUNY, Stonybrook, "Dynamical System Seminar"

JULY 1993

AUGUST 1993

AT&T,

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

SEPTEMBER 1993

OCTOBER 1993

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

APRIL 1994

Stony

MAY
1994

JUNE
1994

Montgomery College, Takoma Park, NSF Teachers Institute.

NOVEMBER
1994

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

JANUARY
1995

FEBRUARY
1995

MARCH
1995

MAY 1995

UMBC,
National Math Honor Society (Induction Lecture)

JUNE
1995

AUGUST 1995

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

OCTOBER 1995

NOVEMBER 1995

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

MARCH
1996

California Institute of Technology, 15th Annual Western States Mathematical Physics Mtg.

APRIL 1996

JUNE
1996

AUGUST 1996

OCTOBER
1996

NOVEMBER 1996

Math
Association of

National

Stony
Brook Dynamical Seminar, Stony

DECEMBER 1996

JANUARY 1997

Dynamics Days--

FEBRUARY 1997

MURI Mtg., Virginia Tech,

MARCH 1997

International Conference on Order
and Chaos,

APRIL 1997

AMS Meeting, Lawrence Berkeley National Labs, CA (presented poster).

MAY 1997

SIAM Meeting, Snowbird,

JUNE 1997

NSF/CMPS
Conference,

NICHD
Meeting on Fetal development,

JULY 1997

Conference on Nonlinear Phenomena in
Dynamical Systems,

AUGUST 1997

MURI Meeting, Virginia Tech.,

SEPTEMBER 1997

IMA Annual Program Workshop,

OCTOBER 1997

GA
Tech, Topology Mtg.,

NOVEMBER 1997

Howard Univ.,

IMA Annual Program Workshop,

DECEMBER 1997

Weizmann Institute,

Rehoboth 3 lectures

JANUARY 1998

AMS
Meeting,

Spring
Topology Mtg.,

Computational
Science Initiative, OER,

FEBRUARY 1998

International Winter School, Max
Planck Institute,

MARCH
1998

Penn
State/UMCP Dynamics Mtg.,

APRIL 1998

JULY 1998

Conference
on Dynamical Systems :

Max
Planck Institute,

SEPTEMBER 1998

Conference on Computational Physics,

OCTOBER 1998

MURI
Meeting, Virginia Tech,

NOVEMBER 1998

Courant
Institute of Mathematical Science, Nonlinear Dynamics Meeting,

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,

APS Centennial,

APRIL 1999

MAY 1999

SIAM Meeting, Snowbird,

JULY 1999

Dynamics Days Asia-Pacific,

The Chinese

The Advanced Crane Technology
Workshop,

Pacific Institute for the Math.
Sciences,

Summer Institute on Smooth Ergodic
Theory and Applications,

AUGUST 1999

OCTOBER
1999

Fall
Leadership Forum, Sandia National Laboratories,

Conference
on Chaos in Physics,

JANUARY 2000

Conference
on Differential Equations,

Joint
SIAM Meeting with AMS and MAA,

MARCH 2000

International Conference on
Fundamental Sciences: Math/Theoretical Physics,

APRIL 2000

British Applied Math Colloquium
2000,

JUNE
2000

Dynamics Days

AUGUST 2000

OCTOBER 2000

International
Workshop on Chaos & Nonlinear Dynamics,

Dynamical
Systems Workshop,

JANUARY 2001

Dynamics Days 2001,

MARCH 2001

Spring Topology and Dynamical
Systems Conference,

MAY 2001

Recomb Satellite Meeting on DNA
Sequence Assembly,

SIAM
Conf. on Applications of Dynamical Systems, Snowbird,

Symposium
in Real Analysis XXV,

AUGUST 2001

Conley Index at

OCTOBER 2001

IMA Workshop #2,

NOVEMBER 2001

DECEMBER 2001

Fields Institute meeting on computational biology

Fields Institute on computational challenges in dynamical systems

JANUARY
2002

Dynamics Days 2002,

MAY 2002

RECOMB 2002,

JUNE 2002

Geometric Theory of Dynamical
Systems Conference,

Session in Honor of Andrzej Lasota,
in

17^{th} Summer Topology
Conference,

AUGUST 2002

New
Directions in Dynamical Systems,

OCTOBER 2002

Dynamical Systems Workshop 2002,

NOVEMBER 2002

“Learning About Reality from
Observation,”

“Better Assembly of Genomes” The Institute for Genomic Research (TIGR)

“Chaos and Prediction,” NSA,

JANUARY 2003

APRIL 2003

Midwest Dynamical Systems meeting,

Japan Prize ceremonies lecture

UC Berkeley math colloquium

MAY 2003

Recomb Satellite Meeting in Stanford on Genome Assembly

SIAM Snowbird Utah Nonlinear Dynamics meeting – 3 30 minute lectures

AUGUST 2003

OCTOBER 2003

Math Colloquium (with JT Halbert) which was also a talk in the Penn State Dynamics meeting.

NOVEMBER 2003

Dyn. System and Its Appl. to Biology, at Nat. Cent. for
Theo. Sciences (CTS)

DECEMBER 2003

Bifurcation Theory and Spatio-Temporal Pattern Formation at the Fields Institute

JANUARY 2004

Dynamics Days Duke-Univ of NC

Amer.
Math. Soc. annual meeting,

MARCH 2004

NIST

APRIL 2004