Comparison between covariant and orthogonal Lyapunov vectors

Abstract

Two sets of vectors, covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (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 with CLVs. For general dissipative systems with nonhyperbolic dynamics the long-wavelength 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 microreversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs, and hyperbolicity under changes of coordinate are discussed in appendices.

Symmetry properties of orthogonal and covariant Lyapunov vectors and their exponents

“…This does not apply to the GS-exponents which suffer from the additional constraint of orthogonality. It has been shown recently by Yang and Radons that covariant vectors may be easily transformed from one coordinate system to another, and the same is true for the covariant finite-time and local Lyapunov exponents [4] We shall demonstrate this property for the chaotic pendulum below by transforming from the Cartesian representation to a polar coordinate system.…”

“…Eq. (25) clearly shows that the local exponents differ for different coordinate systems [4]. To test the last relation, we show by the smooth (red) lines in Fig.…”

“…Although formulated in terms of a particular coordinate transformation, the expressions (23) and (24) are completely general [4]. For our numerical work we use reduced units for which the mass m, the rest length R, and the gravitational acceleration g are unity.…”

“…Practical schemes for the computation of covariant vectors have been developed only recently [11,27]. A few studies of their properties for various systems have appeared since then [4,13,14,15,16,17,18,19,20]. In the following sections we shall study these properties for two simple symplectic systems, the chaotic spring pendulum and the Hénon-Heiles system.…”

“…Various such sets have been introduced in connection with the algorithms for the computation of the Lyapunov exponents. The features of these sets and of their associated exponents are well known to mathematicians and theoretical physicists [1,2,3,4]. In view of an increasing number of applications to ever more sophisticated physical systems, it seems worthwhile to become familiar with the properties of these tangent-space objects for some simple models.…”