A numerical study of the topology of hyperbolic manifolds supporting Arnold diffusion in a priori unstable systems.

Description

In this paper we study the Arnold diffusion along a normally hyperbolic invariant manifold in a model of a priori unstable system. Using numerical methods we detect global and local properties of the stable and unstable manifolds of the invariant manifold, and we compare them with the diffusion properties. Specifically, we introduce a new definition of Arnold diffusion which is adapted to the numerical investigation of the problem, and we show that the numerically computed stable and unstable manifolds indeed support this kind of Arnold diffusion. We also show that the global topology of the stable and unstable manifolds has a transition when the Melnikov approximation loses its accuracy. The 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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