Limit cycles of the generalized mixed Rayleigh-Lienard oscillator

The limit cycles of the generalized mixed Rayleigh-Lienard oscillator are investigated by using the averaging method and singularity theory. The differential equation of system contains highly non-linear terms both in the damping and the restoring coefficients, which results in a highly co-dimensional averaged equation. Applying the singularity theory to the averaged (bifurcation) equation with lower co-dimension shows that two basic modes of the bifurcation diagram correspond to the sub- and super-critical Hopf bifurcations, respectively. All other bifurcation diagrams are generated from these two basic ones through hysteresis bifurcations. Based on the singularity theory approach results, the number of limit cycles and the number of possible bifurcation diagrams with different topological structures (i.e. the bifurcation modes) of the bifurcation equation with higher co-dimension are determined by an inductive method. Analysis suggests that the number of bifurcation modes increase exponentially as the degree of nonlinear terms is increased.