Comparison of covariant and orthogonal Lyapunov vectors

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.

References in corpus (3)

• Characterizing dynamics with covariant Lyapunov vectors
• Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices
• Time-oscillating Lyapunov modes and auto-correlation functions for quasi-one-dimensional systems

Cited by in corpus (10)

• Hyperbolic decoupling of tangent space and effective dimension of dissipative systems
• Covariant Lyapunov Vectors of a Quasi-geostrophic Baroclinic Model: Analysis of Instabilities and Feedbacks
• Jacobian deformation ellipsoid and Lyapunov stability analysis revisited
• Statistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow
• Covariant hydrodynamic Lyapunov modes and strong stochasticity threshold in Hamiltonian lattices
• Backward and covariant Lyapunov vectors and exponents for hard disk systems with a steady heat current
• Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems
• Hyperbolic Covariant Coherent Structures in two dimensional flows
• Using Covariant Lyapunov Vectors to Quantify High Dimensional Chaos with a Conservation Law
• Periodic eigendecomposition and its application to Kuramoto-Sivashinsky system