Arnold diffusion in multidimensional a priori unstable Hamiltonian systems

Abstract

We study the Arnold diffusion in a priori unstable near-integrable systems in a neighbourhood of a resonance of low order. We consider a nonautonomous near-integrable Hamiltonian system with n + 1/2 degrees of freedom, n ≥ 2. Let the Hamilton function H of depend on the parameter ε, for ε = 0 the system is integrable and has a homoclinic asymptotic manifold Γ. Our main result is that for small generic perturbation in an ε-neighborhood of Γ there exist trajectories the projections of which on the space of actions cross the resonance. By "generic perturbations" we mean an open dense set in the space of C r -smooth functions d dε ε=0 H, r = r 0 , r 0 + 1, . . . , ∞, ω. Combination of this result with results of [44] answers the main questions on the Arnold diffusion in a priori unstable case: the diffusion takes place for generic perturbation, diffusion trajectories can go along any smooth curve in the action space with average velocity of order ε/| log ε|.

Gevrey genericity of Arnold diffusion in a priori unstable Hamiltonian systems

“…More precisely, the authors in [32][33][34] defined the so-called scattering map which accounts for the outer dynamics along homoclinic orbits, and overcame the large gap problem by incorporating in the transition chain new invariant objects, like secondary tori and the stable and unstable manifolds of lower dimensional tori; in [73], the author geometrically defined the so-called separatrix map near the NHIC, then he showed in [74] the existence of diffusion by making full use of the dynamics of this map, and even estimated the optimal diffusion speed of order ε/|log ε| (see also [8]). Moreover, for the case of a priori unstable Hamiltonians with higher degrees of freedom, similar results have also been obtained by variational or geometric methods in [3,25,36,37,46,59,75].…”

“…It is also worth mentioning that one can establish the genericity not only in the usual sense but also in the sense of Mañé. Besides, we also believe that our results could be obtained by geometric tools, such as the scattering maps developed in [30,[34][35][36], or the separatrix maps in [37,74,75].…”