Theory of stochastic transitions in area preserving maps

TL;DR

This paper analyzes the nonlinear mechanisms behind stochastic transitions in area-preserving maps, focusing on how invariant sets destabilize and lead to the destruction of KAM curves, advancing understanding of chaos onset.

Contribution

It provides a detailed analysis of nonlinear phenomena causing stochastic transitions and the destabilization of invariant sets in the standard map, extending previous conjectures.

Findings

Identification of nonlinear destabilization mechanisms

Link between invariant set destruction and KAM curve breakdown

Enhanced understanding of chaos onset in area-preserving maps

Abstract

A famous aspect of discrete dynamical systems defined by area-preserving maps is the physical interpretation of stochastic transitions occurring locally which manifest themselves through the destruction of invariant KAM curves and the local or global onset of chaos. Despite numerous previous investigations (see in particular Chirikov, Greene, Percival, Escande and Doveil and MacKay) based on different approaches, several aspects of the phenomenon still escape a complete understanding and a rigorous description. In particular Greene's approach is based on several conjectures, one of which is that the stochastic transition leading to the destruction of the last KAM curve in the standard map is due the linear destabilization of the elliptic points belonging to a peculiar family of invariants sets {I(m,n)} (rational iterates) having rational winding numbers and associated to the last KAM curve. Purpose of this work is to analyze the nonlinear phenomena leading to the stochastic transition in the standard map and their effect on the destabilization of the invariant sets associated to the KAM curves, leading, ultimately, to the destruction of the KAM curves themselves.