Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

Chaotic dynamics of the van der Pol-Duffing oscillator subjected to periodic external and parametric excitations with delayed feedbacks are investigated both analytically and numerically in this manuscript. With the Melnikov method, the critical value of chaos arising from homoclinic or heteroclinic intersections is derived analytically. The feature of the critical curves separating chaotic and non-chaotic regions on the excitation frequency and the time delay is investigated analytically in detail. The monotonicity of the critical value to the excitation frequency and time delay is obtained rigorously. It is presented that there may exist a special frequency for this system. With this frequency, chaos in the sense of Melnikov may not occur for any excitation amplitudes. There also exists a uncontrollable time delay with which chaos always occurs for this system. Numerical simulations are carried out to verify the chaos threshold obtained by the analytical method.