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

Abstract

We derive the bifurcation set for a not previously considered threeparametric Bogdanov-Takens unfolding, showing that it is possible express its vector field as two different perturbed cubic Hamiltonians. By using several firstorder 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 structures of periodic orbits.As an application of these results, we study a family of 3D memristor oscillators, for which the characteristic function of the memristor is a cubic polynomial. We show that these systems have an infinity number of invariant manifolds, and by adding one parameter that stratifies the 3D dynamics of the family, it is shown that the dynamics in each stratum is topologically equivalent to a representant of the above unfolding. Also, based upon the bifurcation set obtained, we show the existence of closed surfaces in the 3D state space which are foliated by periodic orbits. Finally, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role played by the existence of invariant manifolds.

On the Double-Zero Bifurcation of Jerk Systems

“…We notice that the above scenario is similar to that which takes place for Bogdanov's normal form (7) (see [2]). As is pointed out in [2], ".…”

“…For example, the analysis of mathematical models of an internally constrained planar beam equipped with a lumped visco-elastic device and loaded by a follower force [3] or of a non-linear cantilever beam that is externally damped and made of a visco-elastic material [4] reveals among other solutions the existence of a double-zero bifurcation. Oscillators and electronic circuits are modeled by differential equations, and in some cases, they experience a double-zero bifurcation (see, e.g., [5][6][7]). Also, a double-zero bifurcation appears in some chemical reactions (see, e.g., [8,9]) and in fluid mechanics (see, e.g., [10,11]).…”