Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913090603319296 |
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| author | Ashwin, Peter Postlethwaite, Claire Sieber, Jan |
| author_facet | Ashwin, Peter Postlethwaite, Claire Sieber, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03557 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases Ashwin, Peter Postlethwaite, Claire Sieber, Jan Dynamical Systems Adaptation and Self-Organizing Systems 37G10 34C24 37M20 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. |
| title | Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases |
| topic | Dynamical Systems Adaptation and Self-Organizing Systems 37G10 34C24 37M20 |
| url | https://arxiv.org/abs/2605.03557 |