Shilnikov Homoclinic Bifurcation of Mixed-Mode Oscillations

TL;DR

This paper establishes the existence of Shilnikov homoclinic orbits in the Koper model, a prototype for mixed-mode oscillations in slow-fast systems, using continuation and invariant manifold analysis.

Contribution

It is the first to locate and analyze Shilnikov homoclinic orbits in the Koper model, linking them to mixed-mode oscillations and complex bifurcation structures.

Findings

Homoclinic orbits found in a larger family of systems

Parameter continuation reveals crossing with the Koper system

Complex mixed-mode oscillations characterized by return maps

Abstract

The Koper model is a three-dimensional vector field that was developed to study complex electrochemical oscillations arising in a diffusion process. Koper and Gaspard described paradoxical dynamics in the model: they discovered complicated, chaotic behavior consistent with a homoclinic orbit of Shil'nikov type, but were unable to locate the orbit itself. The Koper model has since served as a prototype to study the emergence of mixed-mode oscillations (MMOs) in slow-fast systems, but only in this paper is the existence of these elusive homoclinic orbits established. They are found first in a larger family that has been used to study singular Hopf bifurcation in multiple time scale systems with two slow variables and one fast variable. A curve of parameters with homoclinic orbits in this larger family is obtained by continuation and shown to cross the submanifold of the Koper system. The strategy used to compute the homoclinic orbits is based upon systematic investigation of intersections of invariant manifolds in this system with multiple time scales. Both canards and folded nodes are multiple time scale phenomena encountered in the analysis. Suitably chosen cross-sections and return maps illustrate the complexity of the resulting MMOs and yield a modified geometric model from the one Shil'nikov used to study spiraling homoclinic bifurcations.