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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

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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
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:
DateEvent
December 2013Published
7 November 2013Available
13 August 2013Accepted
30 January 2013Submitted
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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