Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points

TL;DR

This paper proves the existence of stable periodic orbits and chaotic sets in fast-slow systems with Bogdanov-Takens bifurcations using geometric singular perturbation and blow-up methods, linking the flow near fold points to Painlevé equations.

Contribution

It introduces a novel application of the blow-up method to analyze flow near Bogdanov-Takens fold points in fast-slow systems, revealing the structure of the slow manifold.

Findings

Existence of stable periodic orbits in the system

Presence of chaotic invariant sets

Extension of the slow manifold along Painlevé transcendents

Abstract

The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast-slow type having Bogdanov-Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov-Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronqu\'{e}e solution of the first Painlev\'{e} equation in the blow-up space.