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| LIMIT CYCLES OF THE GENERALIZED POLYNOMIAL LIÉNARD DIFFERENTIAL SYSTEMS |
| Amel Boulfoul,Amar Makhlouf |
| (Dept. of Math., 20 August 1955 University, BP26, El Hadaiek 21000, Skikda. Algeria;Dept. of Math., LMA Laboratory, Badji Mokhtar University, BP12, El Hadjar 23000, Annaba. Algeria) |
| DOI： |
| Abstract: |
| Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Li\'enard differential systems \begin{equation*} \begin{cases}
\dot{x} = y+ \epsilon h_{l}^{1}(x)+ \epsilon^{2} h_{l}^{2}(x), \\[5pt]
\dot{y}= -x- \epsilon (f_{n}^{1}(x) y^{2p+1}+ g_{m}^{1}(x))+\epsilon^{2} (f_{n}^{2}(x)y^{2p+1}+ g_{m}^{2}(x)),
\end{cases}
\end{equation*}
which bifurcate from the periodic orbits of the linear center $\dot{x}=y, \; \dot{y}=-x$, where $\epsilon$ is a small parameter. The polynomials $h_{l}^{1}$ and $h_{l}^{2}$ have degree $l$; $f_{n}^{1}$ and $f_{n}^{2}$ have degree $n$; and $g_{m}^{1},\; g_{m}^{2} $ have degree $m$. $p\in \mathbb{N}$ and $[\cdot]$ denotes the integer part function. |
| Key words: limit cycle; periodic orbit; Li\'enard differential system; averaging theory |