On numerical approaches to the analysis of topology of the phase space for dynamical integrability

TL;DR

This paper develops numerical methods, grounded in KAM theory and stochastic analysis, to identify regions of integrability in parameter spaces of dynamical systems with few degrees of freedom.

Contribution

It introduces a general numerical approach to detect obstructions to integrability and localize integrable regions, extending previous methods with new theoretical insights.

Findings

Method successfully identifies non-integrable regions.

Localization of integrability regions in parameter space.

Applications demonstrated on mechanical systems.

Abstract

In this paper we consider the possibility to use numerical simulations for a computer assisted analysis of integrability of dynamical systems. We formulate a rather general method of recovering the obstruction to integrability for the systems with a small number of degrees of freedom. We generalize this method using the results of KAM theory and stochastic approaches to the families of parameter depending systems. This permits the localization of possible integrability regions in the parameter space. We give some examples of application of this approach to dynamical systems having mechanical origin.