Bifurcation set for a disregarded Bogdanov-Takens unfolding: Application to 3D cubic memristor oscillators

Abstract

Motivated by the dynamical analysis of certain memristor-based oscillators, in this paper we derive the bifurcation set for a three-parametric Bogdanov-Takens unfolding that has not been previously considered in the literature (the saddle-focus-saddle case). By using several first-order Melnikov functions, we obtain for the first time analytical approximations for the bifurcation curves corresponding to homoclinic and heteroclinic connections, which along with the curves associated to local bifurcations organize the parametric regions with different qualitative phase planes. Our interest is the study of a family of 3D memristor oscillators, whose memristor characteristic function is a cubic polynomial. We show that these systems have a first integral; thus, after reducing the problem in one dimension, we can take advantage of the bifurcation set previously obtained. For a certain parameter region, the existence of closed surfaces completely foliated by periodic orbits in the original three-dimensional setting is shown. Additionally, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role played by the invariant manifolds associated to the involved first integral.