Comparison of covariant and orthogonal Lyapunov vectors

TL;DR

This paper compares covariant and orthogonal Lyapunov vectors in chaotic systems, showing their similarities and differences, especially regarding hydrodynamic Lyapunov modes, and explores how these vectors influence the understanding of system stability.

Contribution

The study provides a detailed comparison of CLVs and OLVs, demonstrating their effects on hydrodynamic Lyapunov modes and system classification in Hamiltonian and dissipative systems.

Findings

HLMs survive when using CLVs in both Hamiltonian and dissipative systems.

The universality class classification based on dispersion relations holds for CLVs.

The significance of HLMs varies with hyperbolicity and system type when replacing OLVs with CLVs.

Abstract

Two sets of vectors, covariant and orthogonal Lyapunov vectors (CLVs/OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover the previous classification of two universality classes works for CLVs as well, i.e. the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs by CLVs. For general dissipative systems with nonhyperbolic dynamics the long wave length structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the micro- reversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs and hyperbolicity under changes of coordinate are discussed in appendices.