### On the maximum number of limit cycles of a planar differential system

Document Type : Research Paper

Authors

Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria

Abstract

In this work, we are interested in the study of the limit cycles of a perturbed differential system in  $\mathbb{R}^2$, given as follows
$\left\{ \begin{array}{l} \dot{x}=y, \\ \dot{y}=-x-\varepsilon (1+\sin ^{m}(\theta ))\psi (x,y),% \end{array}% \right.$
where $\varepsilon$ is small enough, $m$ is a non-negative integer, $\tan (\theta )=y/x$, and $\psi (x,y)$ is a real polynomial of degree $n\geq1$. We use the averaging theory of first-order to provide an upper bound for the maximum number of limit cycles. In the end, we present some numerical examples to illustrate the theoretical results.

Keywords