The number of limit cycle bifurcation diagrams for the generalized mixed Rayleigh–Liénard oscillator

Abstract

This paper investigates the generalized mixed Rayleigh–Liénard oscillator with highly nonlinear terms. Not restrict to the number of limit cycles, this analysis considers mainly the number of limit cycle bifurcation diagrams of the system. First, the singularity theory approach is applied to the first-order averaged approximation of the system with lower-order nonlinear terms to reveal all possible bifurcation diagrams. By summarizing the generating rule and structural distinction of different bifurcation diagrams, a numerical procedure is then developed. Calculation suggests that the number of bifurcation diagrams increase very fast as the order of nonlinear terms. Lastly, numerical simulations are adopted to approve the analytical results.