Complex bursting dynamics in a Rayleigh–Liénard oscillator

In this paper, the intricate bursting oscillations in a Rayleigh–Liénard oscillator induced by parametric and external slow-varying excitations are proposed. By treating the slow-varying excitations as the generalized state variables, an autonomous system is produced. We identify symmetric bursting oscillations of four distinct types. By simultaneously overlapping the equilibrium branches and the transformed phase portraits and using the fast-slow analysis method, the generation principles of four bursting patterns are disclosed. To explore the parameter qualities associated with the existence of the heteroclinic and homoclinic bifurcations, the Melnikov method is utilized. In addition, we describe the Hopf delay generation mechanism and how the asymptotic theory is used to figure out the delay interval. Furthermore, the precision of the results is demonstrated using the numerical simulations.