Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices

Abstract

The spatiotemporal dynamics of Lyapunov vectors ͑LVs͒ in spatially extended chaotic systems is studied by means of coupled-map lattices. We determine intrinsic length scales and spatiotemporal correlations of LVs corresponding to the leading unstable directions by translating the problem to the language of scale-invariant growing surfaces. We find that the so-called characteristic LVs exhibit spatial localization, strong clustering around given spatiotemporal loci, and remarkable dynamic scaling properties of the corresponding surfaces. In contrast, the commonly used backward LVs ͑obtained through Gram-Schmidt orthogonalization͒ spread all over the system and do not exhibit dynamic scaling due to artifacts in the dynamical correlations by construction. Lyapunov exponents ͑LEs͒ measure the exponential separation ͑or convergence͒ of nearby trajectories and provide the most common tool to characterize spatiotemporal chaos ͑STC͒ ͓1-3͔. Not only exponential separation rates but also spatial correlations are crucial to deal with predictability questions in extended systems ͓4͔. Basically, ͑almost͒ any initial infinitesimal perturbation evolves in time asymptotically aligning along the most unstable direction, i.e., the main Lyapunov vector ͑LV͒. In extended systems, the main LV is found to be localized in space at any given time. However, due to spatial homogeneity, all directions in tangent space are actually equivalent and the localization center keeps moving all over the system. This is known as "dynamic localization" of the main LV ͓5,6͔.In the last decade, there has been some progress in the study of STC with tools borrowed from nonequilibrium statistical physics. In particular, the evolution of perturbations in spatially extended chaotic systems has been shown to be described by multiplicative Langevin-type equations ͓5-8͔. In many cases ͓6͔, the dynamics of perturbations can be expressed in terms of the prototypical stochastic surface growth equation of Kardar-Parisi-Zhang ͑KPZ͒, ‫ץ‬ t h = ‫ץ͑‬ x h͒ 2 + ‫ץ‬ xx h + , where is a white noise ͓9͔. In this Rapid Communication we study the spatiotemporal structure of STC encoded by the LVs. We show that a family of vectors-that we shall call characteristic LVs-carry important information about the real-space structure, localization properties, and space-time correlations. These properties are disclosed only after a logarithmic transformation, so that each LV is mapped into a rough surface. Our results greatly strengthen the link between STC and certain nonequilibrium surface growth models. We also find that the most widely used orthogonal LVs-that appear in the standard GramSchmidt procedure to compute the LEs ͓10,11͔-lack many of these features due to the construction procedure and, therefore, have much less physical significance.We illustrate our results by numerical simulations of coupled-map lattices ͑CMLs͒, which are a prototype of spatially extended dynamical systems exhibiting chaos. The main advantage of CMLs, as compared with e.g., partial differential equat...

Covariant Lyapunov vectors

“…The situation has drastically changed a few years ago, when efficient algorithms have been developed [9,10,11], so that many scientists are now aware that CLVs can offer information on the local geometric structure of chaotic attractors, as opposed to LEs, which are powerful but global quantities. Several papers appeared, where CLVs have been successfully employed to better understand many aspects of chaotic dynamics [12,13,14,15,16,17,18,19,20,21,22]. Some of the relevant questions are touched in this Special Issue.…”

Characteristic Lyapunov vectors in chaotic time-delayed systems

“…1(b). This 'replication property' is a highly nontrivial phenomenon that was originally discovered to occur in chaotic extended dissipative systems [23,24]. We also emphasize that characteristic LVs exhibit a tendency to clusterize, contrary to backward LVs whose localization sites are scattered due to the imposed orthogonality.…”

“…This does not occur all the time but in an intermittent manner. Interestingly, these structural features-namely, replication and clustering-have recently been reported to occur generically for LVs in chaotic spatially extended systems [23][24][25]. This deepens in the analogy between DDSs and systems with extensive chaos in one dimension, which happens to hold even at the level of non-leading LVs.…”

“…The fact that the LV surfaces obey scaling laws and that systems with spatiotemporal chaos could be divided into a few universality classes, according to the scaling of the associated surfaces, has implications in our understanding of chaos in extended systems from a statistical physics point of view. Moreover, the generic replication and clustering of characteristic LVs along the main vector direction in extended systems [23][24][25] (also observed here for time-delay systems, see previous section) makes it even more interesting to determine the universality class of the main LV, since this is expected to provide a great deal of information about the structure of space and time correlations of the LV corresponding to leading as well as sub-leading unstable directions.…”

“…Regarding chaotic systems, in many of them the main LV surface belongs to the KPZ class. This includes coupled-map lattices, coupled symplectic maps, the complex Ginzburg-Landau model, Lorenz 96 model, among others [12,23,24,28]. Only LVs of anharmonic Hamiltonian lattices are known [29] to exhibit correlations that are clearly inconsistent with KPZ exponents (α > α KP Z = 1/2).…”