A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems

TL;DR

This paper rigorously verifies a geometric diffusion mechanism in a specific class of unstable Hamiltonian systems, providing explicit conditions and detailed constructions to demonstrate the existence of diffusing orbits.

Contribution

It offers explicit, verifiable conditions for diffusion in a priori unstable Hamiltonian systems and clarifies the geometric mechanism through detailed construction of the scattering map.

Findings

Explicit conditions for diffusing orbits are derived.

The scattering map construction is fully described.

The geometric mechanism of diffusion is clarified and simplified.

Abstract

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold