Diffusion along chains of normally hyperbolic cylinders

28 Aug 2017  ·  Gidea Marian, Marco Jean-Pierre ·

The present paper is part of a series of articles dedicated to the existence of Arnold diffusion for cusp-residual perturbations of Tonelli Hamiltonians on $\mathbb{A}^3$. Our goal here is to construct an abstract geometric framework that can be used to prove the existence of diffusing orbits in the so-called a priori stable setting, once the preliminary geometric reductions are preformed... Our framework also applies, rather directly, to the a priori unstable setting. The main geometric objects of interest are $3$-dimensional normally hyperbolic invariant cylinders with boundary, which in particular admit well-defined stable and unstable manifolds. These enable us to define, in our setting, chains of cylinders, i.e., finite, ordered families of cylinders in which each cylinder admits homoclinic connections, and any two consecutive elements in the family admit heteroclinic connections. Our main result is the existence of diffusing orbits drifting along such chains, under precise conditions on the dynamics on the cylinders, and on their homoclinic and heteroclinic structure. read more

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Dynamical Systems Mathematical Physics Mathematical Physics Symplectic Geometry Chaotic Dynamics