The Occurrence of Zero-Hopf Bifurcation in a Generalized Sprott A System

• Autor(a) principal: Messias, Marcelo [UNESP]
• Data de Publicação: 2020
• Outros Autores: Reinol, Alisson C.
• Tipo de documento: Artigo de conferência
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://dx.doi.org/10.1007/978-3-030-34713-0_16; http://hdl.handle.net/11449/228898
• topic: Invariant sphere; Invariant torus; Linearly stable periodic orbit; Sprott A system; Zero-Hopf bifurcation

From the normal form of polynomial differential systems in R3 having a sphere as invariant algebraic surface, we obtain a class of quadratic systems depending on ten real parameters, which encompasses the well-known Sprott A system. For this reason, we call them generalized Sprott A systems. In this paper, we study the dynamics and bifurcations of these systems as the parameters are varied. We prove that, for certain parameter values, the z-axis is a line of equilibria, the origin is a non-isolated zero-Hopf equilibrium point, and the phase space is foliated by concentric invariant spheres. By using the averaging theory we prove that a small linearly stable periodic orbit bifurcates from the zero-Hopf equilibrium point at the origin. Finally, we numerically show the existence of nested invariant tori around the bifurcating periodic orbit.