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arXiv · 2605.03557

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

Peter Ashwin·Claire Postlethwaite·Jan Sieber
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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.

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Peter Ashwin, Claire Postlethwaite, Jan Sieber. 2026-05-05. Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases. https://arxiv.org/abs/2605.03557

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