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Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems |
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| Authors: | Joana G. Freire Jason A.C. Gallas |
| Affiliation: | a Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil b Centro de Estruturas Lineares e Combinatórias, Faculdade de Ciências, Universidade de Lisboa, 1649-003 Lisboa, Portugal c Departamento de Física, Universidade Federal da Paraiba, 58051-970 João Pessoa, Brazil |
| Abstract: | We report a systematic two-parameter study of the organization of mixed-mode oscillations and period-adding sequences observed in an extended Bonhoeffer-van der Pol and in a FitzHugh-Nagumo oscillator. For both systems, we construct isospike diagrams and show that the number of spikes of their periodic oscillations are organized in a remarkable hierarchical way, forming a Stern-Brocot tree. The Stern-Brocot tree is more general than the Farey tree. We conjecture the Stern-Brocot tree to also underlie the hierarchical structure of periodic oscillations of other systems supporting mixed-mode oscillations. |
| Keywords: | Mixed-mode oscillations Stern-Brocot tree Farey tree Period-adding Periodicity hub Phase diagram |
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