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.


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Volume 13, Issue 1
March 2022
Pages 1462-1478
  • Receive Date: 04 April 2021
  • Accept Date: 12 September 2021