Maximum number of limit cycles of a second order differential system

Abstract

In this paper, we study the limit cycles of a perturbed differential in $\mathbb{R} ^2$, given by \begin{equation*} \left\{ \begin{array}{ccl} \overset{.}{x} &=& y ,\\ \overset{.}{y} &=& -x-\epsilon (1+\sin^n (\theta) \cos^m (\theta))H(x,y), \end{array} \right. \end{equation*} where $\epsilon$ is a small parameter, $m$ and $n$ are non-negative integers, $\tan(\theta)=y/x$, and $P(x,y)$ is a real polynomial of degree $n \geq 1$. Using Averaging theory of first order we provide an upper bound for the maximum number of limit cycles. Also, we provide some examples to confirm and illustrate our results.