Zero-Hopf Bifurcations in Three-Dimensional Chaotic Systems with One Stable Equilibrium

Author(s)

Ibre, Jaume ^[1] ; Messias, Marcelo ^[2] ; Reinol, Alisson de Carvalho ^[3]

Affiliation

^[1] Univ Autonoma Barcelona UAB, Dept Matemat, Barcelona 08193, Catalonia - Spain

^[2] Univ Estadual Paulista, UNESP, Fac Ciencias & Tecnol, Dept Matemdt & Comp, BR-19060900 P Prudente, SP - Brazil

^[3] Univ Tecnol Fed Parana UTFPR, Dept Acad Matemat, BR-86812460 Apucarana, Parana - Brazil

• Document type: Journal article
• Source: INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS; v. 30, n. 13 OCT 2020.
• Web of Science Citations: 0

Abstract

In {[}Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in R-3 with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in R-3 depending on a real parameter a, which, for a = 1, coincide with the differential systems given by {[}Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value a. We prove that, for a = 0, all the 23 considered systems have a nonisolated zero-Hopi equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero-Hopf bifurcation takes place at this point for a = 0, which leads to the creation of three periodic orbits bifurcating from it for a > 0 small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when a = 1 are obtained by period-doubling route to chaos. (AU)

• FAPESP's process: 19/10269-3 - Ergodic and qualitative theories of dynamical systems II
• Grantee: Claudio Aguinaldo Buzzi
• Support Opportunities: Research Projects - Thematic Grants