Controlling subharmonic resonance and chaos by a fast forcing in a van der Pol–Duffing oscillator with parametrically excited damping

Abstract

A periodically damped van der Pol oscillator is analysed with emphasis on the formation of sub harmonic resonance and the corresponding transition of its motion to the chaotic domain. The slow fast separation technique is used to analytically ascertain the domain of such resonances by the removal of secular terms. The same method is then used in conjunction of multiple scale technique to identify the existence of Hopf bifurcation and the generation of limit cycle. Furthermore the Hamiltonian nature of the system has been utilized in the Melnikov function to determine the threshold of the parameter values for the inset of chaos. Over and above the change in the hysteresis structure due to the periodic damping is explicitly determined. In the chaotic regime the characterization is done with the help of Lyapunov exponent, Poincare map and phase space attractor structure.