Celso Grebogi

CRISES, SUDDEN CHANGES IN CHAOTIC ATTRACTORS, AND TRANSIENT CHAOS

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 presents 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 destruction or creations of chaotic attractors and their basins are due to crises.

The control of chaos: Theory and applications

Abstract Control of chaos refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior. We review the major ideas involved in the control of chaos, and present in detail two methods: the Ott–Grebogi–Yorke (OGY) method and the adaptive method. We also discuss a series of relevant issues connected with chaos control, such as the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions, and point out applications for controlling fractal basin boundaries. In short, we describe procedures for stabilizing desired chaotic orbits embedded in a chaotic attractor and discuss the issues of communicating with chaos by controlling symbolic sequences and of synchronizing chaotic systems. Finally, we give a review of relevant experimental applications of these ideas and techniques.

Strange attractors that are not chaotic

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 word 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 attractore 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 typw 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 observed to be converted to a periodic or chaotic orbit. Thus we conjecture that, in general, continuous time systems (“flows”) 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.

Fractal basin boundaries

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 structures 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.

Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics

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.

Final state sensitivity: An obstruction to predictability

Abstract It is shown that nonlinear system 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.

Do numerical orbits of chaotic dynamical processes represent true orbits

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.

Estimating correlation dimension from a chaotic time series: when does plateau onset occur?

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

Basin boundary metamorphoses: changes in accessible boundary orbits

Abstract Basin boundaries sometimes undergo sudden metamophoses. These metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal, or else can cause a fractal, or else can 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 boundarys accessible orbit.

Bifurcation to chaotic scattering

Abstract We investigate a novel type of bifurcation to chaos which occurs in the context of chaotic scattering. In chaotic scattering the deflection angle versus impact parameter is singular on a set of impact parameters which is fractal. This behavior is caused by the presence of a chaotic invariant set of unstable bounded orbits. In the bifurcation considered here the chaotic set arises abruptly (in a sense to be discussed) as the particle energy E decreases from above a critical value Em, to below. We call this transition an abrupt bifurcation to fully developed chaotic scattering. Numerical computation of the dimension, d, of the chaotic set shows that, in agreement with theoretical prediction, d ≈ 1/In[(Em - E)−1] near the abrupt bifurcatio