A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems

TL;DR

This paper derives a second order expansion of the separatrix map for trigonometric perturbations in unstable systems, enabling better understanding of stochastic diffusion in nearly integrable deterministic systems.

Contribution

It provides a novel second order expansion of the separatrix map specifically for trigonometric perturbations, advancing the analysis of stochastic behavior in such systems.

Findings

Enhanced understanding of stochastic diffusion in nearly integrable systems

Application of the expansion to describe diffusive behavior

Connection to previous studies on separatrix maps and perturbations

Abstract

In this paper we study a so-called separatrix map introduced by Zaslaskii-Filonenko [ZF68] and studied by Treschev and Piftankin [Tre98, Tre02, Pif06, PT07]. We derive a second order expansion of this map for trigonometric perturbations. In [CK15, GK15], and [KZZ15], applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.