Reversal of period doubling, multistability and symmetry breaking aspects for a system composed of a van der pol oscillator coupled to a duffing oscillator

Abstract

Owing to recently published works, the issues of multistability and symmetry breaking can be listed amongst the most followed ongoing research topics in nonlinear science. In this contribution, we consider the dynamics of a system composed of a van der Pol oscillator linearly coupled to a Duffing oscillator (Han, 2000; Kengne et al., 2012). Mention that coupled attractors of different types serve as convenient models for real world systems such as electromechanical, biological, physical, or economic systems. We analyze how the explicit symmetry break modifies the phase space location and nature of equilibrium points of the coupled system, the topology and number of competing attractors, the bifurcation modes, and the shape of the basins of attraction. These investigations are executed by resorting to classical nonlinear tools such as basins of attraction, phase portraits, plots of 1D and 2D largest Lyapunov exponent diagrams, and 1D bifurcation diagrams as well. We report intricate dynamical features such as critical transitions, hysteresis, the coexistence of (symmetric or asymmetric) bubbles of bifurcation and the occurrence of multiple coexisting dynamics (i.e. two, three, four or five coexisting attractors) resulting from the variation of both initial states and parameters of the coupled system.