Geography of resonances and Arnold diffusion ina prioriunstable Hamiltonian systems

Abstract

Abstract. In the present paper we consider the case of a general C r+2 perturbation, for r large enough, of an a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom, and we provide explicit conditions on it, which turn out to be C 2 generic and are verifiable in concrete examples, which guarantee the existence of Arnold diffusion. This is a generalization of the result in Delshams et al., Mem. Amer. Math. Soc., 2006, where it was considered the case of a perturbation with a finite number of harmonics in the angular variables.The method of proof is based on a careful analysis of the geography of resonances created by a generic perturbation and it contains a deep quantitative description of the invariant objects generated by the resonances therein. The scattering map is used as an essential tool to construct transition chains of objects of different topology. The combination of quantitative expressions for both the geography of resonances and the scattering map provides, in a natural way, explicit computable conditions for instability.

Arnold diffusion for a complete family of perturbations

“…By the geometric properties of the scattering map (it is an exact symplectic map [DLS08]) we have, see [DH09] and [DH11], that the scattering map has the explicit form…”

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…It is convenient to consider the lift of the original map so that ϕ runs the whole real axis and the functions p, q and F 0 + f − v are periodic in ϕ. So, in the analysis of the operator Q given by (27), we assume v ∈ R 2 .…”

“…A system of this type is called a priori unstable. The drift of orbits along the cylinder has been actively studied in the last decade [3][4][5][7][8][9]12,[18][19][20]22,27,28,[31][32][33][34]70,87,88], including the problem of genericity of this phenomenon and instability times. It should be noted that the Arnold diffusion can be much faster in this case.…”

Explaining Thermodynamic-Like Behavior in Terms of Epsilon-Ergodicity

“…First, Arnold diffusion has been proven to exist in many concrete examples (see, e.g., Arnold 1964;Berti, Biasco, and Bolle 2003;Mather 2004;Delshams and Huguet 2009). Furthermore, numerical studies confirm the existence of an Arnold web for arbitrarily small perturbations of integrable Hamiltonian systems.…”