Multiple limit cycles of some strongly nonlinear Liénard–Van der Pol oscillator

In this paper, we investigate limit cycles of some Liénard–Van der Pol oscillator; the system has a double homoclinic loop and two cuspidal loops if the damping effect has vanished. By the related Melnikov function theory and bifurcation theories, the limit cycles near the singular circle and center with their distribution are found. The number of limit cycles obtained also reveals that some recent results on the lower bounds of the maximal number of limit cycles bifurcated from this kind of systems can be improved.