Lyapunov Instabilities of Extended Systems

Yang, Hongliu; Radons, Günter

doi:10.1007/3-540-33541-2_18

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…which are related via a non-invertible transformation v = u x . Numerical simulations show that the two forms have the identical hyperbolicity and details will be shown elsewhere [20].…”

Section: Appendix A: Asymptotic and Finite Time Lyapunov Exponentsmentioning

confidence: 89%

Comparison between covariant and orthogonal Lyapunov vectors

Yang

Radons

2010

Phys. Rev. E

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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.

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“…which are related via a non-invertible transformation v = u x . Numerical simulations show that the two forms have the identical hyperbolicity and details will be shown elsewhere [20].…”

Section: Appendix A: Asymptotic and Finite Time Lyapunov Exponentsmentioning

confidence: 89%

Comparison between covariant and orthogonal Lyapunov vectors

Yang

Radons

2010

Phys. Rev. E

Self Cite

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“…In section 8, we go beyond the Lyapunov analysis to probe the geometric structures of the IM via the projection method in order to obtain a direct relation between the identified physical modes and the IM of PDEs. Details of these studies and our previous results on the Lennard-Jones system can be found in [49][50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 98%

Hydrodynamic Lyapunov modes and effective degrees of freedom of extended systems

Yang

Radons

2013

J. Phys. A: Math. Theor.

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This work reviews the authors’ contributions to the Lyapunov analysis of extended dynamical systems. Hydrodynamic Lyapunov modes (HLMs), the special Lyapunov vectors (LVs) associated with near-zero Lyapunov exponents exhibiting long wavelength structures and slow oscillations, have recently been observed in many extended systems with continuous symmetry, such as hard sphere systems, dynamic XY models, Lennard-Jones fluids, coupled map lattices and partial differential equations. They are of potential importance for the connection between nonlinear dynamics and its coarse-grained description. In the first part of this paper, we review our recent results on Lyapunov modes in an extended system, which includes the universality of HLMs in Hamiltonian and dissipative systems, the condition for the appearance of significant or ‘vague’ modes and the appearance of branch splitting in the Lyapunov spectra of diatomic Hamiltonian systems. In particular, most results on HLMs obtained via orthogonal LVs have been checked by using covariant LVs. An emerging picture from these studies is that HLMs in Hamiltonian systems can be viewed as the generalization of normal modes of harmonic systems to nonlinear, chaotic systems. The second part is devoted to the hyperbolicity and the effective degrees of freedom of dissipative partial differential equations. By using covariant LVs, a hyperbolic separation between two sets of LVs was found in these infinite-dimensional systems. The finite set of mutually entangled LVs, named ‘physical modes’, was conjectured to span a local linear approximation of the inertial manifold, a finite-dimensional smooth manifold containing the physically relevant dynamics of the original PDE system. Going beyond the Lyapunov analysis, a projection method was proposed to probe the geometric structures of the inertial manifold, which enables us to provide a direct relation between the physical modes and the inertial manifold of dissipative PDEs. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.

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Lyapunov Instabilities of Extended Systems

Cited by 2 publications

References 26 publications

Comparison between covariant and orthogonal Lyapunov vectors

Comparison between covariant and orthogonal Lyapunov vectors

Hydrodynamic Lyapunov modes and effective degrees of freedom of extended systems

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