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

Document Type : Research Paper


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


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
\dot{x}=y, \\
\dot{y}=-x-\varepsilon (1+\sin ^{m}(\theta ))\psi (x,y),%
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.