On the study of limit cycles of the generalized Rayleigh-Lienard oscillator

摘要

The Hopf bifurcation, saddle connection loop bifurcation and Poincare bifurcation of the generalized Rayleigh-Lienard oscillator X + aX + 2bX(3) + epsilon(c(3) + c(2)X(2) + c(1)X(4) + c(4)X(2)) X = 0 are studied. It is proved that for the case a < 0, b > 0 the system has at most six limit cycles bifurcated from Hopf bifurcation or has at least seven limit cycles bifurcated from the double homoclinic loop. For the case a > 0, b < 0 the system has at most three limit cycles bifurcated from Hopf bifurcation or has three limit cycles bifurcated from the heteroclinic loop.