Complex Periodic Bursting Structures in the Rayleigh–van der Pol–Duffing Oscillator

In the present paper, complex bursting patterns caused by the coupling effect of different frequency scales in the Rayleigh–van der Pol–Duffing oscillator (RVDPDO) driven by the external excitation term are presented theoretically. Seven different kinds of bursting, i.e., bursting of compound “Homoclinic/Homoclinic” mode via “Homoclinic/Homoclinic” hysteresis loop, bursting of compound “fold/Homoclinic-Homoclinic/Hopf” mode via “fold/Homoclinic” hysteresis loop, bursting of compound “fold/Homoclinic-Hopf/Hopf” mode via “fold/Homoclinic” hysteresis loop, bursting of “fold/Homoclinic” mode via “fold/Homoclinic” hysteresis loop, bursting of “fold/Hopf” mode via “fold/fold” hysteresis loop, bursting of “Hopf/Hopf” mode via “fold/fold” hysteresis loop and bursting of “fold/fold” mode are studied by using the phase diagram, time domain signal analysis, phase portrait superposition analysis and slow-fast analysis. With the help of the Melnikov method, the parameter properties related to the beingness of the Homoclinic and Heteroclinic bifurcations chaos of the periodic excitations are investigated. Then, by acting the external forcing term as a slowly changing state variable, the stability and bifurcation characteristics of the generalized autonomous system are given, which performs a major part in the interpretative principles of different bursting patterns. This paper aims to show the sensitivity of dynamical characteristics of RVDPDO to the variation of parameter μ and decide how the choice of the parameters influences the manifold of RVDPDO during the repetitive spiking states. Finally, the validity of the research is tested and verified by the numerical simulations.