Nassima Debz, Amel Boulfoul, Abdelhak Berkane
abstract:
In this work, we study the number of limit cycles which can bifurcate from
periodic orbits of the linear center $\dot{x}=-y$, $\dot{y}=x$ of generalized
polynomial Kukles systems of the form
$$\dot{x}=-y+l(x)\,y^{2\alpha}, \quad
\dot{y}=x-f(x)\,y^{2\alpha}-g(x)\,y^{2\alpha+1}-h(x)\,y^{2\alpha+2}-d_{0}\,y^{2\alpha+3},$$
where
\begin{gather*}
l(x)=\epsilon l_{1}(x)+\epsilon^{2}l_{2}(x), \quad f(x)=\epsilon
f_{1}(x)+\epsilon^{2}f_{2}(x), \\
g(x)=\epsilon g_{1}(x)+\epsilon^{2}g_{2}(x), \;\; h(x)=\epsilon
h_{1}(x)+\epsilon^{2}h_{2}(x) \;\;\text{and}\;\;
d_{0}=\epsilon d^{1}_{0}+\epsilon^{2} d^{2}_{0}.
\end{gather*}
$l_{k}(x)$, $f_{k}(x)$, $g_{k}(x)$ and $h_{k}(x)$ have degree $m$, $n_{1}$,
$n_{2}$ and $n_{3}$, respectively, $d^{k}_{0}\neq 0$ is a real number for each
$k=1,2$, $\alpha$ is a positive integer and $\epsilon$ is a small parameter. The
main tool of this study is the averaging method of the first and second order.
Mathematics Subject Classification: 34C29, 34C25, 47H11
Key words and phrases: Limit cycle, averaging theory, Kukles systems