Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor

In this work we investigate the dynamical behaviors of Van der Pol–Duffing circuit (ADVP) with parallel resistor. The model is described by a continuous-time three dimensional autonomous system. The stability conditions of the equilibria are analyzed. The existence of periodic solutions and their stabilities about the node equilibrium point of the system are studied by using Hopf's theorem and Hsü and Kazarinoff theorem. Lyapunov spectrum is calculated for the proposed system. Adaptive synchronization using backstepping design is applied successfully to the system. Chaotic behaviors and the efficiency of the synchronization method are verified by numerical simulations.