Zero-Hopf bifurcation in a Chua system

Author(s)

Euzebio, Rodrigo D. ; Llibre, Jaume

• Document type: Journal article
• Source: NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS; v. 37, p. 31-40, OCT 2017.
• Web of Science Citations: 3

Abstract

A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are +/- wi, not equal 0 and 0. In general for a such equilibrium there is no theory for knowing when it bifurcates some small-amplitude limit cycles moving the parameters of the system. Here we study the zero-Hopf bifurcation using the averaging theory. We apply this theory to a Chua system depending on 6 parameters, but the way followed for studying the zero-Hopf bifurcation can be applied to any other differential system in dimension 3 or higher. In this paper first we show that there are three 4-parameter families of Chua systems exhibiting a zero-Hopf equilibrium. After, by using the averaging theory, we provide sufficient conditions for the bifurcation of limit cycles from these families of zero-Hopf equilibria. From one family we can prove that 1 limit cycle bifurcates, and from the other two families we can prove that 1, 2 or 3 limit cycles bifurcate simultaneously. (C) 2017 Elsevier Ltd. All rights reserved. (AU)

• FAPESP's process: 10/18015-6 - A study for minimal sets in non-smooth systems in dimensions two and three
• Grantee: Rodrigo Donizete Euzébio
• Support Opportunities: Scholarships in Brazil - Doctorate
• FAPESP's process: 13/25828-1 - Minimal sets and invariant manifolds in piecewise smooth systems
• Grantee: Rodrigo Donizete Euzébio
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 12/05635-1 - Study of families of periodic orbits and their bifurcations in differential equations of finite dimension
• Grantee: Rodrigo Donizete Euzébio
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate