Transcritical and zero-Hopf bifurcations in the Genesio system
In this paper we study the existence of transcritical and zero-Hopf bifurcations of the third-order ordinary differential equation x⃛ + ax¨ + bx˙ + cx- x2= 0 , called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differentia...
| Autores: | , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2017 |
| País: | Brasil |
| Recursos: | Universidade Estadual Paulista (UNESP) |
| Repositório: | Repositório Institucional da UNESP |
| Idioma: | inglês |
| OAI Identifier: | oai:repositorio.unesp.br:11449/174011 |
| Acesso em linha: | http://dx.doi.org/10.1007/s11071-016-3259-2 http://hdl.handle.net/11449/174011 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Averaging theory Genesio system Transcritical bifurcation Zero-Hopf Bifurcation |
| Resumo: | In this paper we study the existence of transcritical and zero-Hopf bifurcations of the third-order ordinary differential equation x⃛ + ax¨ + bx˙ + cx- x2= 0 , called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differential equation as a first-order differential system in R3 we prove: first that the system exhibits a transcritical bifurcation at the equilibrium point located at the origin of coordinates when c= 0 and the parameters (a, b) are in the set { (a, b) ∈ R2: b≠ 0 } \ { (0 , b) ∈ R2: b> 0 } , and second that the system has a zero-Hopf bifurcation also at the equilibrium point located at the origin when a= c= 0 and b> 0. |
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