Maximum number of limit cycles for generalized Liénard polynomial differential systems

Berbache, Aziza ; Bendjeddou, Ahmed ; Benadouane, Sabah

Mathematica Bohemica, Tome 146 (2021) no. 2, pp. 151-165.

Résumé

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.

Détail

DOI : 10.21136/MB.2020.0134-18

Classification : 34C07, 34C23, 37G15
Keywords: polynomial differential system; limit cycle; averaging theory