Dynamics, control and symmetry breaking aspects of a modified van der Pol–Duffing oscillator, and its analog circuit implementation

Abstract

This article focuses on the dynamics of a modified van der Pol–Duffing circuit (MVDPD hereafter) (Fotsin and Woafo in Chaos Solitons and Fractals 24(5):1363–1371, 2005) whose symmetry is explicitly broken with the presence an offset term. When ignoring offset terms, the system displays an exact symmetry which is reflected in the location of the equilibrium points, the attractor topologies and the attraction basins shapes as well. In this mode of operation, the system displays typical behaviors such as period doubling sequences; spontaneous symmetry breaking, symmetry recovering, and multistability involving several pairs of mutually symmetric attractors. In the presence of offset terms, the MVDPD circuit is non-symmetric and more complex nonlinear phenomena arise such as parallel bifurcation branches, coexisting multiple (i.e. two, three, four or five) asymmetric attractors, and crises. It should be noted that for each case of multistability discussed in this work, a hidden attractor (period-1 limit cycle) coexists with self-excited others. To the best of our knowledge, the coexistence of five attractors (symmetrical or asymmetrical), one of which is hidden has not yet been reported in the MVDPD circuit and thus deserves dissemination. PSpice simulation investigations based on the implementation of the MVDPD confirm the theoretical predictions.