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Dependence of Hamiltonian chaos on perturbation structure. (English) Zbl 0925.70164

Summary: We considered a Hamiltonian dynamical system consisting of a steady wave and a perturbation wave and studied the dependence of spatial patterns of chaos on the perturbation structure (i.e., the wave numbers of the perturbation wave). The system came from the passive wave mixing and transport problem. In order to investigate this dependence, we first did some simple mixing experiments with initially a small blob and calculated the correlation dimensions. Secondly we used Lyapunov exponents to identify the chaotic regions and the invariant tori and computed the histograms or PDFs (probability distribution functions) to characterize the Hamiltonian chaos for different perturbation structure. We found that this dependence was very complicated and the complexity increases with the perturbation structure. This dynamical system became more chaotic with increase in the wave numbers. The patterns of the Hamiltonian chaos for various perturbation structures were presented. The spatial pattern of chaos on the isentropic surface of the atmosphere was given. Implications of the results of the chaotic wave mixing and transport in climate dynamics, atmospheric chemistry, aeronomy and large scale dynamics of geophysical fluid flows were briefly discussed.

MSC:

70H05 Hamilton’s equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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