Qualitative analysis, chaos and coexisting attractors in an asymmetric four-well ϕ 8 -generalized Liénard oscillator driven by parametric and external excitations

In this paper, we study the qualitative dynamical analysis, routes to chaos and the coexistence of attractors in a four-well φ 8 -generalized Li´enard oscillator under external and parametric excitations. The local analysis of the autonomous system reveals saddles, nodes, spirals or centers for appropriate choice of stiffness and damping coefficients. The existence of a Hopf bifurcation is proved during the stability analysis of the equilibrium points. The routes to chaos and the prediction of coexisting attractors have been investigated numerically by using the fourth order Runge-Kutta algorithm. The bifurcation structures obtained show that the system displays a rich variety of bifurcation phenomena, such as symmetry breaking, symmetry restoring, period-doubling, period windows, period-m bubbles, reverse period windows, antimonotonicity, intermittency, quasiperiodic, and chaos. In addition, remerging chaotic band attractors and remarkable routes to chaos occur in the system. Further, it is found that the system presents various coexistence of two attractors as well as the monostability and bistability phenomena. On the other hand, for large amplitude of the parametric excitation and with ω = 1, the coexistence of asymmetric periodic bursting oscillations of different topologies takes place in the system. It has also been shown numerically that for appropriate values of system parameters and initial conditions, the presented system can exhibit up to five types of coexisting multiple attractors.