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Title: Three Time-Scales In An Extended Bonhoeffer–Van Der Pol Oscillator
Authors: DE MAESSCHALCK, Peter 
Popovic, Nikola
KUTAFINA, Ekaterina 
Issue Date: 2014
Source: 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.;;
Keywords: Bonhoeffer-van der Pol oscillator; mixed-mode oscillations; canards; geometric singular perturbation theory; blow-up technique
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ISSN: 1040-7294
e-ISSN: 1572-9222
DOI: 10.1007/s10884-014-9356-3
ISI #: 000346171800006
Rights: © Springer Science+Business Media New York 2014.
Category: A1
Type: Journal Contribution
Validations: ecoom 2016
Appears in Collections:Research publications

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