existence, uniqueness and other properties of the limit cycle of a generalized van der pol equation

In this article, we study the bifurcation of limit cycles from the linear oscillator $\dot{x}=y$, $\dot{y}=-x$ in the class $$ \dot{x}=y,\quad \dot{y}=-x+\varepsilon y^{p+1}\big(1-x^{2q}\big), $$ where $\varepsilon$ is a small positive parameter tending to 0, $p \in \mathbb{N}_0$ is even and $q \in \mathbb {N}$. We prove that the above differential system, in the global plane where $p \in \mathbb{N}_0$ is even and $q \in \mathbb{N}$, has a unique limit cycle. More specifically, the existence of a limit cycle, which is the main result in this work, is obtained by using the Poincare's method, and the uniqueness can be derived from the work of Sabatini and Villari [6]. We also investigate and some other properties of this unique limit cycle for some special cases of this differential system. Such special cases have been studied by Minorsky [3] and Moremedi et al [4].