Abstract
We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the splitting determinant is exponentially small.
Highlights
In a previous paper [6] we introduced, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to the “splitting of separatrices” and to the “shadowing problem”
We applied our method to the problem of Arnold Diffusion, i.e. topological instability of action variables, for nearly integrable partially isochronous systems
Later on systems with three time scales have been reconsidered for example in [16], [17], [22], [10]
Summary
In a previous paper [6] (see [7]) we introduced, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to the “splitting of separatrices” and to the “shadowing problem”. It is very well suited to deal with the shadowing problem by means of variational techniques because Gμ is nothing but the difference of the values of the Lagrangian action functional associated to the quasi-periodically forced pendulum (2.2) at two solutions, lying respectively on the stable and unstable manifolds Wμs,u(TI0 ), see (2.4). It may shed light on a “non uniform” splitting which would not be given by the splitting determinant, when the variations of Gμ in different directions are of different orders.
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