Analysis of Three-Dimensional Autonomous Van der Pol–Duffing Type Oscillator and Its Synchronization in Bistable Regime

Abstract

This chapter proposes a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator which is designed from a nonautonomous VdPD two-dimensional chaotic oscillator driven by an external periodic source through replacing the external periodic drive source with a direct positive feedback loop. The dynamical behavior of the proposed autonomous VdPD type oscillator is investigated in terms of equilibria and stability, bifurcation diagrams, Lyapunov exponent plots, phase portraits and basin of attraction plots. Some interesting phenomena are found including for instance, period-doubling bifurcation, symmetry recovering and breaking bifurcation, double scroll chaos, bistable one scroll chaos and coexisting attractors. Basin of attraction of coexisting attractors is computed showing that is associated with an unstable equilibrium. So the proposed autonomous VdPD type oscillator belongs to chaotic systems with self-excited attractors. A suitable electronic circuit of the proposed autonomous VdPD type oscillator is designed and its investigations are performed using ORCAD-PSpice software. Orcard-PSpice results show a good agreement with the numerical simulations. Finally, synchronization of identical coupled proposed autonomous VdPD type oscillators in bistable regime is studied using the unidirectional linear feedback methods. It is found from the numerical simulations that the quality of synchronization depends on the coupling coefficient as well as the selection of coupling variables.