Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases

TL;DR

This paper explores the complex dynamics arising from the interaction of two systems with saddle-node bifurcations, revealing multiple bifurcation types and developing numerical methods to track them.

Contribution

It uncovers the rich bifurcation structure in coupled systems with saddle-node bifurcations and introduces numerical continuation techniques for these complex bifurcations.

Findings

Identification of various local and global bifurcations in coupled systems

Discovery of codimension-two bifurcations like SNIC and non-central SNIC

Development of numerical methods for bifurcation tracking

Abstract

The saddle-node bifurcation is the simplest example of a generic bifurcation in smooth ordinary differential equations, and is associated with the creation or destruction of a pair of equilibria. In this paper we examine the unfolding of the dynamics that occur when two generically coupled systems have simultaneous saddle-node bifurcations. We note that four parameters are required to generically unfold the interactions, and the dynamics are surprisingly complicated relative to the simplicity of a single saddle-node bifurcation. In the unfolding, in addition to saddle-node, Hopf and codimension-two local bifurcations, we also find a variety of global bifurcations, including homoclinic, SNIC, SNICeroclinic and non-central SNIC bifurcations. The latter two are codimension-two bifurcations that occur at the termination of a curve of SNIC bifurcations. A further contribution of this work is the development of numerical continuation techniques for the tracking of these codimension-two bifurcations through parameter space.