TY - JOUR

T1 - Diffusion along transition chains of invariant tori and Aubry-Mather sets

AU - Gidea, Marian

AU - Robinson, Clark

PY - 2013/10/1

Y1 - 2013/10/1

N2 - We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus; (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map; (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains (i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence); (iv) the transition chains of tori are interspersed with gaps created by resonances; (v) within each gap there is prescribed a finite collection of Aubry-Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry-Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.

AB - We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus; (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map; (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains (i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence); (iv) the transition chains of tori are interspersed with gaps created by resonances; (v) within each gap there is prescribed a finite collection of Aubry-Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry-Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.

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U2 - 10.1017/S0143385712000363

DO - 10.1017/S0143385712000363

M3 - Article

AN - SCOPUS:84867840551

VL - 33

SP - 1401

EP - 1449

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 5

ER -