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We derive the bifurcation set for a not previously consideredthree-parametric Bogdanov-Takens unfolding, showing that it is possible expressits vector field as two different perturbed cubic Hamiltonians. By usingseveral first-order Melnikov functions, we obtain for the first time analyticalapproximations for the bifurcation curves corresponding to homoclinic andheteroclinic connections, which along with the curves associated to localbifurcations organize the parametric regions with different structures ofperiodic orbits.  As an application of these results, we study a family of 3D memristoroscillators, for which the characteristic function of the memristor is a cubicpolynomial. We show that these systems have an infinity number of invariantmanifolds, and by adding one parameter that stratifies the 3D dynamics of thefamily, it is shown that the dynamics in each stratum is topologicallyequivalent to a representant of the above unfolding. Also, based upon thebifurcation set obtained, we show the existence of closed surfaces in the 3Dstate space which are foliated by periodic orbits. Finally, we clarify somemisconceptions that arise from the numerical simulations of these systems,emphasizing the important role played by the existence of invariant manifolds.

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