Zero-Hopf bifurcation in a Chua system. (October 2017)

• Record Type: Journal Article
• Authors: Euzébio, Rodrigo D.; Llibre, Jaume
• Is Part Of: Nonlinear analysis. Volume 37(2017)
• Page Start: 31
• Page End: 40
• Publication Date: 2017-10
• Subjects: Chua system -- Periodic orbit -- Averaging theory -- Zero Hopf bifurcation; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2017.02.002
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are ± ω i ≠ 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.