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

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          Abstract

          We derive the bifurcation set for a not previously considered three-parametric Bogdanov-Takens unfolding, showing that it is possible express its vector field as two different perturbed cubic Hamiltonians. 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 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.

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          MEMRISTOR OSCILLATORS

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            New features of the software MatContfor bifurcation analysis of dynamical systems

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              Hidden oscillations in nonlinear control systems

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                Author and article information

                Journal
                12 November 2018
                Article
                1811.04862
                031808fc-d7f8-42d0-92be-08adf719304e

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                24 pages, 10 figures, submitted to Nonlinearity
                math.DS

                Differential equations & Dynamical systems
                Differential equations & Dynamical systems

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