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
38th 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(xT) with continuous G but with highly non-unique solutions of initial value problems. As a side issue, this paper contains a short proof of the Tietze Extension Theorem on metric spaces. If g is continuous on a closed set S in a metric space X, then define G = g on S and for x not in S,
G(x) = inf for y in S of {g(y) + d(x,y)/d(x,S) – 1}. Then G is continuous on X.
10. J. A. Yorke, Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc. 22 (1969), 509-512.
Abstract. Assume dx/dt = F(x) is a differential equation on Rn or on a Hilbert space. Assume F satisfies the Lipshitz condition
|| F(x) – F(y) || <= L || x – y || where || . || denotes the Euclidean metric.
Assume p is a periodic orbit with period T. Then T >= 2 pi / L.
1970
1. J. A. Yorke, Asymptotic stability for one-dimensional differential delay-equations, J. Differential Equations 7 (1970), 189-202.
2. J. A. Yorke, A continuous differential equation in Hilbert space without existence, Funkcialaj Ekvacioj 13 (1970), 19-21.
3. J. A. Yorke, Differential inequalities and non-Lipschitz scalar functions, Math. Systems Theory 4 (1970), 140-153.
4. Gerald S. Goodman and J. A. Yorke, Misbehavior of solutions of the differential equation dy/dx = f(x,y) + epsilon, when the right side is discontinuous, Mathematica Scandinavica 27 (1970), 72-76.
Abstract: It is well known that by consideration of the corresponding integral equation, most qualitative theorems concerning initial-value problems for the first order ordinary differential equation dy/dx = f (x,y) can be extended to the case where the right side is no longer continuous. In this note, however, we shall show by example that more than one widely used theorem in the continuous case cannot be so extended, at least not in a form that would preserve its most useful feature, as soon as the right side of the equation fails to be jointly continuous at just a single point, even though it remains bounded and continuous there in each variable separately.
5. A. Strauss and J. A. Yorke, Linear perturbations of ordinary differential equations , Proc. Amer. Math. Soc. 26 (1970), 255-260.
Abstract: We present several results dealing with the problem of the preservation of the stability of a system dx/dt=A(t)x that is subject to linear perturbations B(t)x, or to perturbations dominated by linear ones.
6. J. A. Yorke, A theorem on Lyapunov functions using the second derivative of V, Math. Systems Theory 4 (1970), 40-45.
1971
1. A. Lasota and J. A. Yorke, Oscillatory solutions of a second order ordinary differential Equation, Ann. Polon. Math. 25 (1971), 175-178.
2. J. A. Yorke, Another proof of the Lyapunov convexity theorem, SIAM J. Control (1971), 9 351-353.
Abstract: A new proof of the Liapunov convexity theorem is presented.
3. S. Saperstone and J. A. Yorke, Controllability of linear oscillatory systems using positive controls, SIAM J. Control 9 (1971), 253-272.
Abstract: A linear autonomous control process is considered where the null control is an extreme point of the restraint set S. In the even that S=[0,1] (hence, scalar control) necessary and sufficient conditions are given so that the reachable set from the origin (in phase space) contains the origin as an interior point. For vector-valued controls with each component in [0,1], sufficient conditions are given so that the reachable set from the origin of a nonlinear autonomous control process contains the origin as an interior point.
4. A. Lasota and J. A. Yorke, Bounds for periodic solutions of differential equations in Banach spaces, J. Differential Equations 10 (1971), 83-91.
1972
1. A. Lasota and J. A. Yorke, Existence of solutions of two-point boundary value problems for nonlinear systems, J. Differential Equations 11 (1972), 509-518.
2. J. A. Yorke, The maximum principle and controllability of nonlinear equations, SIAM J. Control 10 (1972), 334-338.
Abstract: The main result proved is that a nonlinear control equation is controllable if a related linear equation is controllable. The result allows the set of control values to be discrete and it is not assumed that small values of the control are available. The methods used are closely related to the Pontryagin maximum principle.
3. S. Grossman and J. A. Yorke, Asymptotic behavior and stability criteria for differential delay equations), J. Differential Equations 12 (1972), 236-255.
4. S. Bernfeld and J. A. Yorke, The behavior of oscillatory solutions of x"(t)+p(t)g(x(t))=0, SIAM J. Math. Anal. 3 (1972), 654-667.
Abstract: Various quantitative properties of oscillatory solutions of the scalar second order nonlinear differential equation are obtained under appropriate hypotheses on p and g. In particular, letting {ti, 0 < ti < ti+1, where ti goes to infinity, be the zeroes of any solution x(t), we obtain inequalities that yield asymptotic behavior on x(t). For example, it is shown that the integral of g(x(ti)) exists and is finite: moreover, assuming an added growth condition on g(x)/x, we have then that the integral of x(t) from 0 to infinity exists and is finite.
1973
1. F. W. Wilson, Jr. and J. A. Yorke, Lyapunov functions and isolating blocks, J. Differential Equations 13 (1973), 106-123.
2. K. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci. 16 (1973), 75-101.
Abstract: At the present time VD is a major national problem. Essentially we are confronted with several epidemics. This paper is devoted to a study of processes of this nature. It is hoped that understanding of the mathematical nature of these processes will help in their control.
3,4. W. London, M.D. and J. A. Yorke, Recurrent outbreaks of measles, chicken pox, and mumps, I. Seasonal variation in contact rates, and II. Systematic differences in contact rates and stochastic effects, Amer. J. Epidemiology 98 (1973), 453-468 and 469-482.
5. A. Lasota and J. A. Yorke, The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations 13 (1973), 1-12.
6. A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488.
Abstract: A class of piecewise continuous, piecewise C1 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 C1 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 Rn 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 Aj , A is a subset of Tk(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, Ti(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 Ti(x) to oscillate chaotically on A for a certain time and eventually escape A. For each measurable set E in A define mk(A) to be the conditional probability that Tk(x) is in E given that x, T1(x), ...,Tk (x) are in A. We show that mk converges to a smooth measure m0 that is independent of the choice of m which we call a “conditionally invariant measure”. One-dimensional examples are stressed.
5. J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Comm. Math. Phys. 67 (1979), 93-108. This paper is reprinted in Russian in a book edited by Sinai and Kolmogorov on strange attractors.
Abstract: This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short-term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value that is the first value for which the system possesses a homoclinic orbit.
6. J. A. Yorke and E. D. Yorke Metastable chaos: The transition to sustained chaotic oscillations in a model of Lorenz, J. Stat. Phys. 21 (1979), 263-277.
Abstract: The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbers r somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence upon r is studied both numerically and (very close to the critical r) analytically.
1980
1. J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. 32 (1980), 177-188.
Abstract: We investigate the dynamical properties of continuous maps of a compact metric space into itself. The notion of chaos is defined as the instability of all trajectories in a set together with the existence of a dense orbit. In particular we show that any map on an interval satisfying a generalized period three condition must have a nontrivial (uncountable) minimal set as well as large chaotic subsets. The nontrivial minimal sets are investigated by lifting to sequence spaces while the chaotic sets are investigated using factors, projections of larger spaces onto smaller ones.
1981
1. A. Lasota and J. A. Yorke, The law of exponential decay for expanding mappings , Rend. Sem. Mat. Univ. Padova 64 (1981), 141-157.
2. K. T. Alligood, J. Mallet-Paret and J. A. Yorke, Families of periodic orbits: Local continuability does not imply global continuability, J. Differential Geom. 16 (1981), 483-492.
1982
1. J. Mallet-Paret and J. A. Yorke, Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differential Equations 43 (1982), 419-450.
Abstract: Poincare observed that for a differential equation dx/dt = f(x, a) depending on a parameter a, each periodic orbit generally lies in a connected family of orbits in (x,a)- space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: defined at certain stationary points. We show that generically there are two types of Hopf bifurcation, those we call sources (K = 1) and sinks (K = -1). Generically if the set Q is bounded in (x, a)-space, and if there is an upper bound for periods of the orbits in Q, the Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter snake of orbits. A snake is a maximal path of orbits that contains no orbits whose orbit index is 0.
2. H. W. Hethcote, J. A. Yorke and A. Nold, Gonorrhea modeling: A comparison of control methods, Math. Biosci. 58 (1982), 93-109.
Abstract: A population dynamics model for a heterogeneous population is used to compare the effectiveness of six prevention methods for gonorrhea involving population screening and contact tracing of selected groups. The population is subdivided according to sex, sexual activity, and symptomatic or asymptomatic infection. For this model contact tracing of certain groups is more effective than general population screening.
3. A. Lasota and J. A. Yorke, Exact dynamical systems and the Frobenius-Perron operator. Trans. Amer. Math. Soc. 273 (1982), 375-384.
Abstract: Conditions are investigated that guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps T on [0,1] for which dT/dx(0) > 1. We assume [0,1] can be broken into intervals on which T is continuous and convex and at the left end of these intervals, T = 0 and dT/dx > 0. Such maps have an invariant absolutely continuous density that is exact.
4. T. Y. Li, M. Misiurewicz, G. Pianigiani and J. A. Yorke, Odd chaos, Phys. Lett. 87A (1982), 271-273.
Abstract: The simplest chaotic dynamical processes arise in models that are maps of an interval into itself. Sometimes chaos can be inferred from a few successive data points without knowing the details of the map. Chaos implies knowledge of initial data is insufficient for accurate long term prediction.
5. T. Y. Li, M. Misiurewicz, G. Pianigiani, and J. A. Yorke, No division implies chaos, Trans. Amer. Math. Soc. 273 (1982), 191-199.
Abstract: Let I be a closed interval in R1 and let f be a continuous map on I. Let x0 in I and xi+1 = f (xi) for i 0. We say there is no division for (x1, x2,...,xn) if there is no a in I such that xj < a for all j even and xj < a for all j odd. The main result of this paper proves the simple statement: no division implies chaos. Also given here are some converse theorems, detailed estimates of the existing periods, and examples that show that, under our conditions, one cannot do any better.
6. C. Grebogi, E. Ott and J. A. Yorke, Chaotic attractors in crisis, Phys. Rev. Lett. 48 (1982), 1507-1510. Announcement of #1983-3.
Abstract: The occurrence of sudden qualitative changes of chaotic (or turbulent) dynamics is discussed and illustrated within the context of the one-dimensional quadratic map. For this case, the chaotic region can suddenly widen or disappear, and the cause and properties of these phenomena are investigated.
1983
1. P. Frederickson, J. L. Kaplan, E. D. Yorke and J. A. Yorke, The Lyapunov dimension of strange attractors, J. Differential Equations 49 (1983), 185-207.
Abstract: Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.
2. J. C. Alexander and J. A. Yorke, On the continuability of periodic orbits of parametrized three dimensional differential equations, J. Differential Equations 49 (1983), 171-184.
3.
C. Grebogi,
Abstract: The occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated. It is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor. We call such collisions crises. Phenomena associated with crises include sudden changes in the size of chaotic attractors, sudden appearances of chaotic attractors (a possible route to chaos), and sudden destructions of chaotic attractors and their basins. This paper present examples illustrating that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs. In particular the phenomenon of chaotic transients is investigated. The examples discussed illustrate crises in progressively higher dimension and include the one-dimensional quadratic map, the (two-dimensional) Henon map, systems of ordinary differential equations in three dimensions and a three-dimensional map. In the case of our study of the three-dimensional map a new route to chaos is proposed that is possible only in invertible maps or flows of dimension at least three or four, respectively. Based on the examples presented the following conjecture is proposed: almost all sudden changes in the size of chaotic attractors and almost all sudden destructions or creations of chaotic attractors and their basins are due to crises.
4.
J. D. Farmer,
Abstract: Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the dimension of the natural measure, and all of the metric dimensions take on a common value, which we call the fractal dimension. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
5. C. Grebogi, E. Ott and J. A. Yorke, Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation, Phys. Rev. Lett. 50 (1983), 935-938, E 51 (1983), 942.
Abstract: A new type of bifurcation to chaos is pointed out and discussed. In this bifurcation two unstable fixed points or periodic orbits are created simultaneously with a strange attractor that has a fractal basin boundary. Chaotic transients associated with the coalescence of the unstable-unstable pair are shown to be extraordinarily long-lived.
6. C. Grebogi, E. Ott and J. A. Yorke, Are three frequency quasiperiodic orbits to be expected in typical nonlinear dynamical systems?, Phys.Rev. Lett. 51 (1983), 339-342. Announcement of #1985-4.
Abstract: The current state of theoretical understanding related to the question posed in the title is incomplete. This paper presents results of numerical experiments that are consistent with a positive answer. These results also bear on the problem of characterizing possible routes to chaos in nonlinear dynamical systems.
7. J. A. Yorke and K. T. Alligood Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull. Amer. Math. Soc. 9 (1983), 319-322. Announcement of #1985-7.
8. C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Final state sensitivity: An obstruction to predictability, Phys. Letters 99A (1983), 415-418.
Abstract: It is shown that nonlinear systems with multiple attractors commonly require very accurate initial conditions for the reliable prediction of final states. A scaling exponent for the final-state-uncertain phase space volume dependence on uncertainty in initial conditions is defined and related to the fractal dimension of basin boundaries.
1984
1. J. L. Kaplan, J. Mallet-Paret and J. A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory and Dyn. Sys. 4 (1984), 261-281.
Abstract: The fractal dimension of an attracting torus Tk in R X Tk is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1.
2. B. Curtis Eaves and J. A. Yorke, Equivalence of surface density and average directional density, Math. Operations Res. 9 (1984), 363-375.
Abstract: The average directional density criteria for evaluating tilings is shown to be equivalent to surface density and valid for random broken paths just as for straight paths.
3. K. T. Alligood and J. A. Yorke, Families of periodic orbits: Virtual periods and global continuability, J. Differential Equations 55 (1984), 59-71.
Abstract: For a differential equation depending on a parameter, there have been numerous investigations of the continuation of periodic orbits as the parameter is varied. Mallet-Paret and Yorke investigated in generic situations how connected components of orbits must terminate. Here we extend the theory to the general case, dropping genericity assumptions.
4. J. C. Alexander and J. A. Yorke, Fat baker's transformations, Ergodic Theory and Dyn. Sys. 4 (1984), 1-23.
Abstract: We investigate a variant of the baker transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated.
5. B. R. Hunt and J. A. Yorke, When all solutions of dx/dt = Σi qi(t)x(t-Ti(t)) oscillate, J. Differential Equations 53 (1984), 139-145.
Abstract: In this paper the long-term behavior of solutions to the equation in the title are examined, where qi(t) and Ti(t) are positive. In particular, it is shown that if liminf sumi = ln Ti(t)qi(t) > 1/ e, all solutions oscillate about 0 infinitely often.
6. A. Lasota, T. Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751-764.
Abstract: We say the operator P on L1 is a Markov operator if (i) Pf >= 0 for f >= 0 and (ii) |Pf| = |f| if f >= 0. It is shown that any Markov operator P has certain spectral decomposition if, for any f in L1 with f = 0 and the norm of f = 1, Pnf converges to f when n goes to infinity, where F is a strongly compact subset of L1. It follows from this decomposition that Pnf is asymptotically periodic for any f in L1.
7. C. Grebogi, E. Ott, S. Pelikan and J. A. Yorke, Strange attractors that are not chaotic, Physica 13D (1984), 261-268.
Abstract: It is shown that in certain types of dynamical systems it is possible to have attractors that are strange but not chaotic. Here we use the work strange to refer to the geometry or shape of the attracting set, while the word chaotic refers to the dynamics of orbits on the attractor (in particular, the exponential divergence of nearby trajectories). We first give examples for which it can be demonstrated that there is a strange