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.

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.

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 condition 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, SIAM Rev. 11 (1969), 236-246.

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.

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 which 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 which is subject to linear perturbations B(t)x, or to perturbations dominated by linear ones.

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

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.

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 precesses 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.

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

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.

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

The techniques used are related to the Poincare-Bendixson method, used in the (y(t), y(t-1) - plane.

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.

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.

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.

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 A_j and let T be a smooth map on Rⁿ with A a subset of T(A). Assume that for each A_j , A is a subset of T ^k(A_j) 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_Adm / dx 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 (x), ....T ^k (x) are in A. We show that m_k converges to a smooth measure m₀ which is independent of the choice of m. 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 which 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.

1980

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

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.

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: an orbit index J and a center index K 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 which 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 which 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 x_i + 1 = f (x_i) for i 0. We say there is no division for (x₁, 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 which 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.

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, E. Ott and J. A. Yorke, Crises, sudden changes in chaotic attractors, and transient chaos, Physica 7D (1983), 181-200. Announcement in #1982-6.

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 which 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, E. Ott and J. A. Yorke, The dimension of chaotic attractors, Physica 7D (1983), 153-180.

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 which 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 which 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 = sum_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 lim inf sum_i=1ⁿ 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 L¹ is a Markov operator if (i) Pf > 0 for f > 0 and (ii) the norm of Pf equals the norm of 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 ||f|| = 1, P ⁿf converges to a single limit g independent of f when n goes to infinity, where F is a strongly compact subset of L¹. It follows from this decomposition that P ⁿf 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 which 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 which model nonlinear oscillators (continuous time) which are externally driven at two incommensurate frequencies. It is then shown that such attractors are persistent under perturbations which preserve the original system type (i.e., there are two incommensurate external driving frequencies). This suggests that, for systems of the type which 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 which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observe to be converted to a periodic or chaotic orbit. Thus we conjecture that, in general, continuous time systems which 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.

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 which 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 which 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 which 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 which 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 reach 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

1986

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

Abstract: Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which 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 which 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 R³. He proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane, and that such parameter values m would be dense in some interval. The fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes. We prove, for an apparently typical situation, that the Newhouse construction yields only a set of parameter values m of measure zero.

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

Abstract: The average lifetime of a chaotic transient versus a system parameter is studied for the case wherein a chaotic attractor is converted into a chaotic transient upon collision with its basin boundary (a crisis). Typically the average lifetime T depends upon the system parameter p via T is proportional [p-p₀]^-g, where p₀ denotes the value of p at the crisis and we call (the critical exponent of the chaotic transient. A theory determining (for two-dimensional maps is developed and compared with numerical experiments. The theory also applies to critical behavior at interior crises.

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 which 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 which 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 which 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 which 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 which 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 d₁, d₂,...,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 d₁, d₂,...,d_N. For example, in our double rotor map it appears that N = 2, and d₁ = 3.0 and d₂ = 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 which 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 which 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.

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 =1/22, 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 delta_x -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 which 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 which 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 which 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, E. Ott and J. A. Yorke, Roundoff-induced periodicity and the correlation dimension of chaotic attractors, Phys. Rev. A 38 (1988), 3688-3692.

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.

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.

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 transition. 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 d₀-k [p-p_I], where d₀ is the dimension at the intermittency transition parameter value p = p_I and k is a scaling constant. Furthermore, for type-I intermittency d₀ < D, while for type-III intermittency d₀ = 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. I. Kan and J. A. Yorke, Antimonotonicity: Concurrent creation and annihilation of periodic orbits, Bull. Amer. Math. Soc. 23 (1990), 469-476. Announcement of #1992-1.

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 which 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. I. Kramer, J. A. Yorke and E. D. Yorke, The AIDS epidemic's influence on New York City's gay sexual contact rate from an analysis of gonorrhea incidence, Math. Comput. Modelling 13 (l990), 21-25.

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 which 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. 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.

4. B. R. 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 which 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 which 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 which 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 Rⁿ, 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 Rⁿ. These results are extended in two other directions. Similar results are proved in the more general case of reconstructions which 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

Abstract: One-parameter families f_a of diffeomorphisms of the Euclidean plane are known to have a complicated bifurcation pattern as a 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_a(x) = ax(1-x), where it is known that periodic orbits are created and never annihilated as a 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 Em_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 which shrinks to the point E₀ as m tends to 0, and for m 0 there may be a stable period 3 orbit which similarly shrinks to E₀ 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 which 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 which 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 which 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.

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: MGOY 1983-8 introduced the uncertainty dimension as a quantitative measure for final state sensitivity in a system. In MGOY 1983-8 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.

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

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

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

Abstract: Theory and examples of attractors with basins which are of positive measure, but contain no open sets, are developed; such basins are called riddled. A theorem is established which 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. R. 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 which hold almost everywhere in the sense of prevalence. For instance, we prove that almost ever C ¹ map on Rⁿ 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: 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 D₂ of the underlying chaotic attractor. We report a rigorous result which implies that, for long enough data sets, the plateau begins when the number of delay coordinates first exceeds D₂. 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 2D₀ (here D₀ denotes the box-counting dimension). For calculating the correlation dimension D₂, m D₂ 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 which 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.

Abstract: Suppose that a dynamical system has a chaotic attractor A with a correlation dimension D₂. 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 D₂ for the attractor in the original full phase space. In this paper we first present rigorous results which state that, for a long enough data string with low enough noise, the plateau onset occurs at m = Ceil (D₂), where Ceil (D₂), standing for ceiling of D₂, is the smallest integer greater than or equal to D₂. 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 (D₂).

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 which 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 interated 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. 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 C ⁴ map on a 3-manifold which has an invariant set with an uncountable number of components, each of which is a pseudocircle. Furthermore, any map which is sufficiently close (in the C ¹-metric) to the constructed map has a similar set.

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 which 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_F 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 E₀ 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 - x² + 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. 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 a C infinitely differentiable diffeomorphism on a 7-manifold which has a compact invariant set such that uncountably many of its connected components are pseudocircles. (Any 7-manifold will suffice.) Furthermore, any diffeomorphism which is sufficiently close (in the C ¹ metric) to the constructed map has a similar invariant set, and the dynamics of the map on the invariant set are chaotic.

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

Abstract: We demonstrate and analyze a bifurcation producing a type of fractal basin boundary which 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 B₁, B₂,..., 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. R. 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., et al, 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 which are consistent with those obtained experimentally by Metcalfe et al.

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

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

9. J. 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 which has an invariant set with an uncountable number of pseudocircle components. Furthermore, any diffeomorphism which 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. R. 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 which 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. R. 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. Kennedy, C. Grebogi and J. A. Yorke, Indecomposable continua in dynamical systems with noise: fluid flow past an array of cylinders, Int.J. Bifurcation & 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 x₀ and an initial time t₀. We discuss when the set of initial points at a time t₀ whose trajectory (x(t), y(t)) is semibounded (i.e., x(t) x₀ 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. R. 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 systems to a chaotic drive is a smooth function of the drive state. We call this differentiable generalize synchronization (DGS). When DGS does not hold, we quantify the degree of nondifferentiability using the Holder exponent. We also discuss the consequences of DGS and give an illustrative numerical example.

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

Abstract: In dynamical systems examples are common in which two or more attractors coexist, and in such cases, the basin boundary is nonempty. When there are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the two remaining basins? Is it possible that all three boundaries of these basins coincide? When this last situation occurs the boundaries have a complicated structure. This phenomenon does occur naturally in simply dynamical systems. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. We introduce the basic notion of a basin cell. A basin cell is a trapping region generated by some well chosen periodic orbit and determines the structure of the corresponding basin. This new notion will play a fundamental role in our main results. We consider diffeomorphisms of a two-dimensional smooth manifold M without boundary, which has at least three basins. A point x in M is a Wada point if every open neighborhood of x has a nonempty intersection with at least three different basins. We call a basin B a Wada basin if every x in the boundary of the closure of B is a Wada point. Assuming B is the basin of a basin cell (generated by a periodic orbits P), we show the B is a Wada basin if the unstable manifold of P intersects at least three basins. This result implies conditions for basins B₁, B₂,..., B_N (N=3) to all have exactly the same boundary.

4. E. Barreto, B. R. 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. R. Hunt and J. A. Yorke Correlation dimension for iterated function systems, Trans. Amer. Math. Soc. (1977), 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 which 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 which 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 which 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. 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 typical smooth function, if the phase space is at least D-dimensional. If m is a Borel probability measure of bounded support in Rⁿ with correlation dimension D, and if k = D, then under almost every continuously differentiable function (almost every in the sense of prevalence) from Rⁿ 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. 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 which 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 D₁ of the attractor, reliable prediction is found to be still possible from most orbit points. In particular, the fraction of state space measure from which prediction fails typically scales as epsilon ^M-D1 for small epsilon where epsilon is the diameter of the neighborhood current state used for prediction.

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

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

Dimension

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 is the information dimension of the attractor), rather than 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.

8. K. T. 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.

1999

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

P-1 G.-C. Yuan and J. A. Yorke, An open set of maps for which every point is absolutely nonshadowable, Proc. Amer. Math. Soc., in press

P-2. J. Kennedy and J. A. Yorke, Dynamical system topology preserved in the presence of noise, accepted by Turkish Journal of Math.

P-3. J. Miller and J. A. Yorke, Finding all periodic orbits of maps using Newton methods: sizes of basins, submitted to Physica D.

P-4. L. Poon, C. Grebogi, T. Sauer and J. A. Yorke, Limits to deterministic modeling, submitted to Nature.

P-5. L. Poon, S. P. Dawson, T. Sauer, C. Grebogi, D. Auerbach and J. A. Yorke, Shadowability of chaotic dynamical systems, submitted to Physica D.

P-6. F. Suppe and J. A. Yorke, Modeling HIV risk for promiscuous homosexual men selecting HIV-negative partners, submitted to Science.

P-7. S. Guharay, B. R. Hunt, J. A. Yorke and O. White, Correlation in gene sequences across the three domains of life, submitted to Phys. Rev. Lett.

P-8. G.-C. Yuan and J. A. Yorke, Collapsing of Chaos in one dimensional maps, submitted to Physica D.

P-9. D. Sweet, E. Ott and J. A. Yorke, Dimension Formula for an atypical chaotic saddle system, in preparation.

P-10. G.-H.Yuan, B. R. Hunt, C. Grebogi, E. Ott and J. A. Yorke, A chaotically forced pendulum: ship cranes at sea, in preparation.

P-11. H. Siegelman, J. A. Yorke and A. Katz, Approximation Algorithms for Real Computation, in preparation..

P-12. J. Kennedy and J. A. Yorke, Topological horseshoes, in preparation

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

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

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

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

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

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

8. J. Curry and J. A. Yorke, A transition from Hopf bifurcation to chaos: Computer experiments with maps in R^2, in The Structure of Attractors in Dynamical Systems, Springer Lecture Notes in Math #668, 48-66: The proceedings of the NSF regional conference in 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, Institute of Mathematics, Kiev (1977), 15-16: Anniversary volume in honor of I. Mitropolsky.

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

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

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

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

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

15. T. Y. Li and J. A. Yorke, A simple reliable numerical algorithm for following homotopy paths, in Analysis and Computation of Fixed Points, Academic Press (1980), 73-91: The proceedings of Math. Res. Center conference at the University of Wisconsin, May 1979.

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

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

18. J. Harrison and J. A. Yorke, Flows on 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, Signapore, 1984), pp. 108-110.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

36. J. Levine, P. Rouchon, G. Yuan, C. Grebogi, B. R. 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.

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

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

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

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

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

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

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

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

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

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

11. N. Nathanson, G. Pianigiani, J. Martin and J. A. Yorke, Requirements for perpetuation and eradication of viruses in populations, in Persistent Viruses, Academic Press, 1978, 76-100: The proceedings of 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 Manhattan in December 1979.

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

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

15. S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, An obstacle to predictability, in Proc. XIIIth Intl Colloq. on Group Theor. 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 Theor. 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. Providence, RI: American Mathematical Society series in Contemporary Mathematics, Vol. 99 (1989), pp. 307-312.

23. J. A. Yorke, Chaos and scientific knowledge, proceedings of conference on "Individuality and Cooperation Action" (ed. Joseph E. Earley), Georgetown University, Washington, DC, April, l991.

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, Springer-Verlag, New York, etc. l991.

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.

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