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Covariant Lyapunov vectors
The recent years have witnessed a growing interest for covariant Lyapunov
vectors (CLVs) which span local intrinsic directions in the phase space of
chaotic systems. Here we review the basic results of ergodic theory, with a
specific reference to the implications of Oseledets' theorem for the properties
of the CLVs. We then present a detailed description of a "dynamical" algorithm
to compute the CLVs and show that it generically converges exponentially in
time. We also discuss its numerical performance and compare it with other
algorithms presented in literature. We finally illustrate how CLVs can be used
to quantify deviations from hyperbolicity with reference to a dissipative
system (a chain of H\'enon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam
chain)
Theory and computation of covariant Lyapunov vectors
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 Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and
by Wolfe and Samelson [Tellus 59A, 2007, 355]. 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.Comment: 21 pages, 5 figure
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