Zero‐Hopf bifurcation in the FitzHugh–Nagumo system |
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| Authors: | Rodrigo D Euzébio Jaume Llibre Claudio Vidal |
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| Institution: | 1. Departament de Matemática, IBILCE, UNESP, Jardim Nazareth, Sao José de Rio Preto, S?o Paulo, Brazil;2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain;3. Departamento de Matemática, Universidad del Bío‐Bío, Concepción, Avda. Collao 1202, Chile |
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| Abstract: | We characterize the values of the parameters for which a zero‐Hopf equilibrium point takes place at the singular points, namely, O (the origin), P+, and P? in the FitzHugh–Nagumo system. We find two two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at the origin is a zero‐Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero‐Hopf equilibrium point O. We prove that there exist three two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at P+ and at P? is a zero‐Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at P+ and P?. Copyright © 2014 John Wiley & Sons, Ltd. |
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| Keywords: | FitzHugh– Nagumo system periodic orbit averaging theory zero‐Hopf bifurcation |
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