Covariant Lyapunov vectors

Abstract

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 Hénon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam chain).

References in corpus (15)

• The large deviation approach to statistical mechanics
• Characterizing dynamics with covariant Lyapunov vectors
• Nonequilibrium fluctuations in a resistor
• Theory and computation of covariant Lyapunov vectors
• Hyperbolicity and the effective dimension of spatially-extended dissipative systems
• Structure of characteristic Lyapunov vectors in spatiotemporal chaos
• Hyperbolic decoupling of tangent space and effective dimension of dissipative systems
• Entropy, Thermostats and Chaotic Hypothesis
• Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices
• Lyapunov analysis captures the collective dynamics of large chaotic systems
• Extensive and Sub-Extensive Chaos in Globally-Coupled Dynamical Systems
• Covariant Lyapunov vectors for rigid disk systems
• Time-reversed symmetry and covariant Lyapunov vectors for simple particle models in and out of thermal equilibrium
• Geometry of inertial manifolds probed via a Lyapunov projection method
• Anomalous transport induced by nonhyperbolicity

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