Characterizing Dynamics with Covariant Lyapunov Vectors

Abstract

A general method to determine covariant Lyapunov vectors in both discrete-and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents. DOI: 10.1103/PhysRevLett.99.130601 PACS numbers: 05.70.Ln, 05.90.+m, 45.70.ÿn, 87.18.Ed Measuring Lyapunov exponents (LEs) is a central issue in the investigation of chaotic dynamical systems because they are intrinsic observables that allow us to quantify a number of different physical properties such as sensitivity to initial conditions, local entropy production and attractor dimension [1]. Moreover, in the context of spatiotemporal chaos, the very existence of a well-defined Lyapunov spectrum in the thermodynamic limit is a proof of the extensivity of chaos [2], and it has been speculated that the small exponents contain information on the ''hydrodynamic'' modes of the dynamics (e.g., see [3] and references therein).In this latter perspective, a growing interest has been devoted not only to the LEs but also to some corresponding vectors, with the motivation that they could contribute to identifying both the real-space structure of collective modes [4] and the regions characterized by stronger or weaker instabilities [5]. However, the only available approach so far is based on the vectors yielded by the standard procedure used to calculate the LEs [6]. This allows us to identify the most expanding subspaces, but has the drawback that these vectors-that we shall call GramSchmidt vectors (GSV) after the procedure used-are, by construction, orthogonal, even where stable and unstable manifolds are nearly tangent. Moreover, GSV are not invariant under time reversal, and they are not covariant; i.e., the GSV at a given phase-space point are not mapped by the linearized dynamics into the GSV of the forward images of this point.While the existence, for invertible dynamics, of a coordinate-independent, local decomposition of phase space into covariant Lyapunov directions -the so-called Oseledec splitting [1]-has been discussed by Ruelle long ago [7], it received almost no attention in the literature, because of the absence of algorithms to practically determine it. In this Letter, we propose an innovative approach based on both forward and backward iterations of the tangent dynamics, which allows determining a set of directions at each point of phase space that are invariant under time reversal and covariant with the dynamics. We argue that, for any invertible dynamical system, the intrinsic tangent-space decomposition introduced by these covariant Lyapunov vectors (CLV) coincides with the Oseledec splitting.As a first important and general application of the CLV...

Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

“…For example, it may be possible to connect our results with experimental measurements using ideas based upon Lagrangian coherent structures [36,37] or computational homology [38]. From a theoretical point of view, our work suggests that it would be interesting to explore the dynamics of the spectrum of Lyapunov vectors using the more recently suggested approach of characteristic Lyapunov vectors that satisfy Oseledec splitting [39,40]. Overall, we anticipate that our results will be useful to those interested in controlling, predicting, and modeling high-dimensional chaotic systems.…”

Collective Lyapunov modes

“…We numerically integrate Equation (2) by the fourth-order Runge-Kutta method with time step 0.1 and compute LEs and CLVs using Ginelli et al's algorithm [24,25]. Most data presented in this section are recorded over a period longer than 10 5 after a transient of length 10 4 or more is discarded.…”

“…This frustrating situation was overcome when Ginelli et al proposed an efficient algorithm to compute covariant Lyapunov vectors (CLVs) in large dynamical systems [24,25]. The CLVs are the vectors spanning the subspaces of the Oseledec decomposition of tangent space [13].…”

Hyperbolic Chaos of Turing Patterns

“…The hyperbolicity implies that there are no tangencies between the stable and unstable manifolds of orbits belonging to the attractor. Occurrence of a tangency is determined by the zero angle between the expanding and contracting tangent subspaces spanned by the corresponding covariant Lyapunov vectors [14]. Following the method for testing hyperbolicity described in [15], we examine the distribution of these angles by considering the orthogonal complement to the contracting subspace, which is normally much less dimensional then the contracting subspace itself.…”