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Interaction of two systems with saddle-node bifurcations on invariant circles : I. Foundations and the mutualistic case
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Baesens, C. and MacKay, Robert S. (2013) Interaction of two systems with saddle-node bifurcations on invariant circles : I. Foundations and the mutualistic case. Nonlinearity, Volume 26 (Number 12). pp. 3043-3076. doi:10.1088/0951-7715/26/12/3043
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Official URL: http://dx.doi.org/10.1088/0951-7715/26/12/3043
Abstract
The saddle-node bifurcation on an invariant circle (SNIC) is one of the codimension-one routes to creation or destruction of a periodic orbit in a continuous-time dynamical system. It governs the transition from resting behaviour to periodic spiking in many class I neurons, for example. Here, as a first step towards theory of networks of such units the effect of weak coupling between two systems with a SNIC is analysed. Two crucial parameters of the coupling are identified, which we call δ1 and δ2. Global bifurcation diagrams are obtained here for the 'mutualistic' case δ1δ2 > 0. According to the parameter regime, there may coexist resting and periodic attractors, and there can be quasiperiodic attractors of torus or cantorus type, making the behaviour of even such a simple system quite non-trivial. In a second paper we will analyse the mixed case δ1δ2 < 0 and summarize the conclusions of this study.
| Item Type: | Journal Article | ||||||||||
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| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | ||||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||||
| Library of Congress Subject Headings (LCSH): | Dynamics, Bifurcation theory | ||||||||||
| Journal or Publication Title: | Nonlinearity | ||||||||||
| Publisher: | Institute of Physics Publishing Ltd. | ||||||||||
| ISSN: | 0951-7715 | ||||||||||
| Official Date: | December 2013 | ||||||||||
| Dates: |
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| Volume: | Volume 26 | ||||||||||
| Number: | Number 12 | ||||||||||
| Number of Pages: | 35 | ||||||||||
| Page Range: | pp. 3043-3076 | ||||||||||
| DOI: | 10.1088/0951-7715/26/12/3043 | ||||||||||
| Status: | Peer Reviewed | ||||||||||
| Publication Status: | Published | ||||||||||
| Access rights to Published version: | Open Access | ||||||||||
| Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||||
| Grant number: | EP/G021163/1 (EPSRC) | ||||||||||
| Open Access Version: |
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