Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion

Abstract

We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+¿j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I¿I¿Rd,f¿TdI¿I¿Rd,f¿Td, p,q¿Rnp,q¿Rn, t¿T1t¿T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I¿I¿RdI¿I¿Rd. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in andPeer ReviewedPostprint (author's final draft

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

“…As an application, we show the existence of diffusing orbits in a large class of nearly integrable a priori unstable Hamiltonian systems that are multi-dimensional both in the center and in the hyperbolic directions. The model below is an extension of those considered in [34,37,42]. Let…”

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…Note also that by (35) we have γ i+1 ≤ M k−i−1 β i+1 . By (28), (34), this gives us that for all k large enough, for every i = 0, . .…”

“…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.…”

Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

“…Nevertheless in [DLS07] it is argued that there exist diffusing orbits -under the assumption that h is polynomial -plus some non-degeneracy assumptions. The key observation is that, under a twist condition, the multiple resonances can be contoured.…”

“…13 The fact that in [DLS06b] one only got control on one of the variables made the calculation of subsequent properties more complicated than what is nowadays possible using geometric theory. See [DLS07]. On the other hand, the calculation based on estimating the change of energy is natural for the purposes of the study of the intersection with KAM toriwhich are given as level sets of the averaged energy.…”