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Theory and computation of covariant Lyapunov vectors. (English) Zbl 1301.37065

Summary: Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by F. Ginelli et al. [“Characterizing dynamics with covariant Lyapunov vectors”, Phys. Rev. Lett. 99, No. 13, Article ID 130601 (2007; doi:10.1103/PhysRevLett.99.130601)] and by C. L. Wolfe and R. M. Samelson [“An efficient method for recovering Lyapunov vectors from singular vectors”, Tellus A 59, No. 3, 355–366 (2007; doi:10.1111/j.1600-0870.2007.00234.x)]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.

MSC:

37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37B25 Stability of topological dynamical systems

Software:

LAPACK; ALGLIB
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Full Text: DOI arXiv

References:

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