Transcritical and zero-Hopf bifurcations in the Genesio system

• Autor(a) principal: Cardin, Pedro Toniol [UNESP]
• Data de Publicação: 2017
• Outros Autores: Llibre, Jaume
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://dx.doi.org/10.1007/s11071-016-3259-2; http://hdl.handle.net/11449/174011
• topic: Averaging theory; Genesio system; Transcritical bifurcation; Zero-Hopf Bifurcation

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.