A proof of Wang–Kooij's conjectures for a cubic Liénard system with a cusp

In this paper the global dynamics of a cubic Liénard system with a cusp is studied to follow Wang and Kooij (1992) [13], who proved that the maximum number of limit cycles is 2 and stated two conjectures about the curves of the cuspidal loop bifurcation and the double limit cycle bifurcation. We give positive answers to those two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness. Finally, associated with previous results we obtain the complete bifurcation diagram and all phase portraits, and demonstrate some numerical examples.