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| Content Provider | Springer Nature Link |
|---|---|
| Author | Kutafina, E. Popović, N. Maesschalck, P. |
| Copyright Year | 2014 |
| 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. (Phys Lett A 374(36):3745–3751, 2010) and by Freire and Gallas (Phys Lett A 375:1097–1103, 2011). 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. (J Appl Dyn Syst 7(2):361–420, 2008) as a canonical form for such systems. Based on results previously obtained in Krupa et al. (2008), 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. (2010). 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 (SIAM J Appl Math 49(2): 331–343, 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. |
| Ending Page | 987 |
| Page Count | 33 |
| Starting Page | 955 |
| File Format | |
| ISSN | 10407294 |
| e-ISSN | 15729222 |
| Journal | Journal of Dynamics and Differential Equations |
| Issue Number | 4 |
| Volume Number | 26 |
| Language | English |
| Publisher | Springer US |
| Publisher Date | 2014-03-18 |
| Publisher Place | Boston |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Bonhoeffer–van der Pol oscillator Canard solutions Bifurcations of limit cycles and periodic orbits Singular perturbations Geometric singular perturbation theory Multiple scale methods Mixed-mode oscillations Canards Singular perturbations, turning point theory, WKB methods Ordinary Differential Equations Relaxation oscillations Blow-up technique Applications of Mathematics Partial Differential Equations |
| Content Type | Text |
| Resource Type | Article |
| Subject | Analysis |
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