| Summary: | 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].
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