ON THE STUDY OF LIMIT CYCLES OF THE GENERALIZED RAYLEIGH–LIENARD OSCILLATOR

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The Hopf bifurcation, saddle connection loop bifurcation and Poincaré bifurcation of the generalized Rayleigh–Liénard oscillator Ẍ+aX+2bX3+ε(c3+c2X2+c1X4+c4Ẋ2)Ẋ=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.