Maximum number of limit cycles for generalized Liénard polynomial differential systems

Abstract

We consider limit cycles of a class of polynomial differential systems of the form \begin{cases} \dot{x}=y, \dot{y}=-x-\varepsilon(g_{21}( x) y^{2\alpha+1} +f_{21}(x) y^{2\beta})-\varepsilon^2(g_{22}( x) y^{2\alpha+1}+f_{22}( x) y^{2\beta}), \end{cases} where $\beta$ and $\alpha$ are positive integers, $g_{2j}$ and $f_{2j}$ have degree $m$ and $n$, respectively, for each $j=1,2$, and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$ using the averaging theory of first and second order.

Keywords

• polynomial differential system
• limit cycle
• averaging theory