Zero-Hopf bifurcation in a Chua system

Abstract

Abstract. A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are ±ωi = 0 and 0. In general for a such equilibrium there is no theory for knowing when from it bifurcates some small-amplitude limit cycle 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 bifurcate, and from the other two families we can prove that 1, 2 or 3 limit cycles bifurcate simultaneously.

Zero-Hopf Bifurcation in the Rössler’s Second System

“…In their work, the maximum number of periodic orbits was three. Furthermore, the first and second orders of averaging theory were used to study the bifurcating periodic orbits of the Fitz-Hugh-Nagumo system and the Chua system by Euzébio et al (Euzébio, 2015) and Llibre and Euzébio (Euzébio, 2017) respectively.…”

Zero-Hopf bifurcation in a 3-D jerk system

“…Then system (9) has the periodic solution of period T ε (u(t, ε), v(t, ε), w(t, ε)) = (r(t, ε) cos θ(t, ε), r(t, ε) sin θ(t, ε), w(t, ε)) , for ε sufficiently small, with (u(0, ε), v(0, ε), w(0, ε)) = (r i , 0, w i ). Now we apply the change of variables (8) to this one and obtain for small ε the periodic solution (X(t, ε), Y (t, ε), Z(t, ε)) of system (7) with the same period, such that (X(0, ε, Y (0, ε), Z(0, ε)) = (w i + r i /δ, r i , 0). Finally, for ε = 0 sufficiently small, system (6) has the periodic solution (x(t, ε), y(t, ε), z(t, ε)) = (εX(θ), εY (θ), εZ(θ)), with (x(0, ε), y(0, ε), z(0, ε)) = ε(w i +r i /δ, r i , 0), and that clearly tends to the origin of coordinates when ε → 0.…”

“…Our analysis, however, will use the averaging theory (see Section 2 for the results on this theory needed for our study). The averaging theory has already been used to study Hopf and zero-Hopf bifurcations in some others differential systems, see for instance [3,5,8,9,14,15,16].…”

Jerk forms dynamics of a Chua’s family and their new unified circuit implementation

“…It could be replaced by other nonlinear functions, such as the staircase functions, polynomial functions, and triangular wave functions, so as to generate various multi-scroll attractors from the original structure of the Chua system within these nonlinear functions [2][3][4][5][6][7][8][9]. Tang et al presented an alternative Chua's circuit, which is, replacing the Chua's diode with a polynomial circuit [6,10] or sine function [2]. These nonlinear circuits are also suitable for the four-order and higher-order Chua system to bring about chaotic attractors.…”

“…Tang et al. presented an alternative Chua’s circuit, which is, replacing the Chua’s diode with a polynomial circuit [6, 10] or sine function [2]. These nonlinear circuits are also suitable for the four‐order and higher‐order Chua system to bring about chaotic attractors.…”