Mechanisms for the Development of Unstable Dimension Variability and the Breakdown of Shadowing in Coupled Chaotic Systems (original) (raw)

Parametric evolution of unstable dimension variability in coupled piecewise-linear chaotic maps

Physical Review E, 2011

In presence of unstable dimension variability numerical solutions of chaotic systems are valid only 8 for short periods of observation. For this reason, analytical results for systems that exhibit this 9 phenomenon are needed. Aiming to go one step further in obtaining such results, we study the 10 parametric evolution of unstable dimension variability in two coupled bungalow maps. Each of 11 these maps presents intervals of linearity that define Markov partitions, which are recovered for the 12 coupled system in the case of synchronization. Using such partitions we find exact results for the 13 onset of unstable dimension variability and for contrast measure, which quantifies the intensity of 14 the phenomenon in terms of the stability of the periodic orbits embedded in the synchronization 15 subspace. 16 Unstable dimension variability (UDV) is a form of non-17 hyperbolicity in which there is no continuous splitting 18 between stable and unstable subspaces along the chaotic 19 invariant set [1]. The variability takes place when the 20 periodic orbits, embedded in the chaotic set, have a dif-21 ferent number of unstable directions. This is a local phe-22 nomenon that can influence the entire phase space, and 23 create complexity in the system [2-4]. Validity of trajec-24 tories generated by chaotic systems that exhibit UDV is 25 guaranteed for short periods [5], which decreases as the 26 intensity of the UDV increases [6, 7]. 27 The intensity of the UDV can be quantified by the em-28 bedded UPOs in a nonhyperbolic attractor [3]. There are 29 efficient computational methods for the analysis of these 30 orbits [8, 9]. However, it is a time-consuming task be-31 cause the number of orbits increases with their period, 32 and in many problems it is necessary to consider very 33 high periods [10-12]. To avoid this problem, one con-34 structs a model so that the UDV occurs in a transversal 35 direction to a hyperbolic attractor. The dynamics in this 36 attractor is well known, and therefore, some analytical re-37 sults can be obtained. This type of construction allows us 38 develop tools in order to shed light on the UDV [13, 14]. 39 Examples of physical problems that can be handled by 40 these tools are: the effect of shadowing in the kicked 41 double-rotor [15, 16], the beginning of the spatial activ-42 ity in the three-waves model [17, 18], transport properties 43 of passive inertial particles incompressible flows [19], and 44 the chaos synchronization in coupled map lattices [20-45 22]. In some cases, the study of periodic orbits embedded 46 in the synchronization subspace allows the determination 47

Unstable dimension variability: A source of nonhyperbolicity in chaotic systems

International Symposium on Physical Design, 1996

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.

Variations in the dimension of chaotic systems

Physics Letters A, 1984

It is usually expected that the number of modes necessary to model turbulence increases with the appropriate control parameter. With the help of a concrete model, we show that this property may be shared by low-dimensional truncations of the Navier-Stokes equations.

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

Physical Review E - PHYS REV E, 2003

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapu...

Chaotic bursting at the onset of unstable dimension variability

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 2002

Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model ͑biased random walk with reflecting barrier͒ is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.

From High Dimensional Chaos to Stable Periodic Orbits: The Structure of Parameter Space

Physical Review Letters, 1997

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. [S0031-9007(97)03367-X] PACS numbers: 05.45.+b 0031-9007͞97͞78(24)͞4561(4)$10.00

Is the dimension of chaotic attractors invariant under coordinate changes?

Journal of Statistical Physics, 1984

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

The natural measure of nonattracting chaotic sets and its representation by unstable periodic orbits

2002

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems.

Loss of Chaos Synchronization through the Sequence of Bifurcations of Saddle Periodic Orbits

Physical Review Letters, 1997

In the work we investigate the bifurcational mechanism of the loss of stability of the synchronous chaotic regime in coupled identical systems. We show that loss of synchronization is a result of the sequence of soft bifurcations of saddle periodic orbits which induce the bubbling and riddling transitions in the system. A bifurcation of a saddle periodic orbit embedded in the chaotic attractor determines the bubbling transition. The phenomenon of riddled basins occurs through a bifurcation of a periodic orbit located outside the symmetric subspace.