Limit Cycles of the Generalized Polynomial Liénard Differential Systems

Abstract

Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Liénard differential systems $$\begin{cases} \dot{x}= y + ϵh^1_l (x) + ϵ^2h^2_l (x),  \\ \dot{y}= −x − ϵ(f^1_n (x)y^{2p+1}+ g^1_m(x)) + ϵ^2 (f^2_n(x)y^{2p+1}+ g^2_m(x)), \end{cases}$$ which bifurcate from the periodic orbits of the linear center $\dot{x} = y,$ $\dot{y}= −x,$ where $ϵ$ is a small parameter. The polynomials $h^1_l$ and $h^2_l$ have degree $l;$ $f^1_n$ and $f^2_n$ have degree $n;$ and $g^1_m,$ $g^2_m$ have degree $m.$ $p ∈ \mathbb{N}$ and [·] denotes the integer part function.