Non-existence, existence, and uniqueness of limit cycles for a generalization of the Van der Pol-Duffing and the Rayleigh-Duffing oscillators

We study the limit cycles of some cubic family of differential equations, containing the well-known Van der Pol-Duffing and Rayleigh-Duffing oscillators. In particular, we characterize for the class of differential systems here studied the non-existence, existence and uniqueness of limit cycles. Moreover we provide their global phase portraits in the Poincaré disk