A numerical study of the topology of hyperbolic manifolds supporting diffusion in a priori unstable systems.
Résumé
Using new numerical methods we detect the topology of hyperbolic manifolds supporting diffusion in the a priori unstable dynamical systems and compare them with the diffusion properties. We measure a spread of the asymptotic manifolds which is significant to explain diffusion. We show that the stable and unstable manifolds have a topological transition when the Melnikov approximation looses its accuracy. This transition is correlated to a change of the law of dependence of the diffusion coefficient on the perturbing parameter. This suggests that the Melnikov approximation is not only a technical tool which allows one to compute accurate approximations of the manifolds at small values of the perturbing parameters, but is related to a dynamical regime.
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