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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/16481

Title: Three Time-Scales In An Extended Bonhoeffer–Van Der Pol Oscillator
Authors: DE MAESSCHALCK, Peter
Popovic, Nikola
KUTAFINA, Ekaterina
Issue Date: 2014
Citation: Journal of Dynamics and Differential Equations, 26 (4), p. 955-987
Abstract: We consider an extended three-dimensional Bonhoeffer-van der Pol oscillator which generalises the planar FitzHugh-Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. and by Freire and Gallas. Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. as a canonical form for such systems. Based on previously obtained results in, we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer-van der Pol oscillator from the point of view of geometric singular perturbation theory, thus complementing the findings reported in Sekikawa et al. In particular, we specify in detail the mixed-mode patterns that are observed upon variation of a bifurcation parameter which is naturally obtained by combining two of the original parameters in the system, and we derive asymptotic estimates for the corresponding parameter intervals. We thereby also disprove a conjecture of Tu (1989), where it was postulated that no stable periodic orbits of mixed-mode type can be observed in an equivalent extension of the Bonhoeffer-van der Pol equations.
Notes: Popovic, N (reprint author), Univ Edinburgh, Sch Math, James Clerk Maxwell Bldg,Kings Bldg,Mayfield Rd, Edinburgh EH9 3JZ, Midlothian, Scotland. Peter.DeMaesschalck@uhasselt.be; Ekaterina.Kutafina@uhasselt.be; Nikola.Popovic@ed.ac.uk
URI: http://hdl.handle.net/1942/16481
DOI: 10.1007/s10884-014-9356-3
ISI #: 000346171800006
ISSN: 1040-7294
Category: A1
Type: Journal Contribution
Validation: ecoom, 2016
Appears in Collections: Research publications

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