A strong form of Arnold diusion for two and a half degrees of freedom

By Vadim Kaloshin, Ke Zhang. 2012

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In the present paper we prove a strong form of Arnold diusion. Let T 2 be the two torus and B 2 be the unit ball around the origin in R 2 . Fix > 0. Our main result says that for a \generic" time-periodic perturbation of an integrable system of two degrees of freedom H0(p) +"H1(;p;t ); 2 T 2 ; p2 B 2 ; t2 T = R=Z; with a strictly convexH0, there exists a -dense orbit ( ;p ;t)(t) in T 2 B 2 T, namely, a -neighborhood of the orbit contains T 2 B 2 T. Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are usage of crumpled normally hyperbolic invariant cylinders from [13], ower and simple normally hyperbolic invariant manifolds from [47] as well as their kissing property at a strong double resonance. This allows us to build a \connected" net of 3-dimensional normally hyperbolic invariant manifolds. To construct diusing orbits along this net we

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c84906a0c90aa9bec1e122af28b1e262db970af8