Chaos and the approach to equilibrium in a discrete sine-Gordon equation. (English) Zbl 0761.58014

Summary: We study the long time dynamics of a Hamiltonian system with many degrees of freedom. We numerically investigate the system’s approach to equipartition of energy when the initial energy is confined to one or a small set of Fourier modes. We find that there is a transition from a low energy regime in which energy does not spread appreciably among the modes to a high energy regime in which the system rapidly approaches equipartition, and that this transition coincides with the onset of chaos in the system, as evidenced by a sharp rise in the largest Lyapunov exponent. For low frequency initial conditions, the critical parameter for this transition is the scaled energy \(E'=(L/N)^ 2E\). Using a generalization of the traditional Chirikov resonance overlap calculation on a three-mode subset of the full system, we predict the onset of widespread chaos and the transition to equipartition on relatively short time scales.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
39A12 Discrete version of topics in analysis
70H05 Hamilton’s equations
Full Text: DOI


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