Limit cycles for a class of polynomial differential systems via averaging theory

Ahmed Bendjeddou ; Aziza Berbache ; Abdelkrim Kina

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 2, pp. 145-159.

Résumé

In this paper, we consider the limit cycles of a class of polynomial differential systems of the form \begin{equation*} \left\{ \begin{array}{l} \dot{x}=y-\varepsilon (g_{11}\left( x\right) y^{2\alpha +1}+f_{11}\left( x\right) y^{2\alpha })-\varepsilon ^{2}(g_{12}\left( x\right) y^{2\alpha +1}+f_{12}\left( x\right) y^{2\alpha }) ,\\ \dot{y}=-x-\varepsilon (g_{21}\left( x\right) y^{2\alpha +1}+f_{21}\left( x\right) y^{2\alpha })-\varepsilon ^{2}(g_{22}\left( x\right) y^{2\alpha +1}+f_{22}\left( x\right) y^{2\alpha }), \end{array} \right. \end{equation*} where $m,n,k,l$ and $\alpha $ are positive integers, $g_{1\kappa }$, $ g_{2\kappa },f_{1\kappa }$ and $f_{2\kappa }$ have degree $n,m,l$ and $k$, respectively for each $\kappa =1,2$, 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

Keywords: averaging theory, Liénard differential systems.
Mots-clés : limit cycles

Bibliographie

[1] J. Alavez-Ramirez, G. Blé, J. Llibre, J. Lopez-Lopez, “On the maximum number of limit cycles of a class of generalized Liénard differential systems”, Int. J. Bifurcation and Chaos, 22 (2012), 1250063 | DOI | MR | Zbl

[2] T.R. Blows, N.G. Lloyd, “The number of small-amplitude limit cycles of Liénard equations”, Math. Proc. Camb. Phil. Soc., 95 (1984), 359–366 | DOI | MR | Zbl

[3] A. Buica, J. Llibre, “Averaging methods for finding periodic orbits via Brouwer degree”, Bull. Sci. Math., 128 (2004), 7–22 | DOI | MR | Zbl

[4] X. Chen, J. Llibre, Z. Zhang, “Sufficient conditions for the existence of at least $n$ or exactly $n$ limit cycles for the Lienard differential systems”, J. Differential Equations, 242 (2007), 11–23 | DOI | MR | Zbl

[5] C.J. Christopher, S. Lynch, “Small-amplitude limti cycle bifurcations for Liénard systems with quadratic or cubic dapimg or restoring forces”, Nonlinearity, 12 (1999), 1099–1112 | DOI | MR | Zbl

[6] W.A. Coppel, “Some quadratic systems with at most one limit cycles”, Dynamics Reported, v. 2, Wiley, New York, 1998, 61–68 | MR

[7] B. Garca, J. Llibre, J.S. Pérez del Rio, “Limit cycles of generalized Liénard polynomial differential systems via averaging theory”, Chaos Solitons Fractals, 62–63 (2014), 1–9 | DOI | MR

[8] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, 1979 | MR

[9] M. Han, P. Yu, Normal forms, Melnikov functions and bifurctions of limit cycles, Applied Mathematical Sciences, 181, Springer, London, 2012 | DOI | MR

[10] J. Li, “Hilbert's 16th problem and bifurcations of planar polynomial vector fields”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13:1 (2003), 47–106 | DOI | MR | Zbl

[11] J. Llibre, A.C. Mereu, M.A. Teixeira, “Limit cycles of generalized polynomial Liénard differential equations”, Math. Proc. Camb. Phil. Soc., 2009, 000, 1 pp. | MR

[12] J. Llibre, C. Valls, “On the number of limit cycles of a class of polynomial differential systems”, Proc. A: R. Soc., 468 (2012), 2347–2360 | DOI | MR | Zbl

[13] J. Llibre, C. Valls, “Limit cycles for a generalization of Liénard polynomial differential systems”, Chaos, Solitons and Fractals, 46 (2013), 65–74 | DOI | MR | Zbl

[14] J. Llibre, C. Valls, “On the number of limit cycles for a generalization of Liénard polynomial differential systems”, Int. J. Bifurcation and Chaos, 23 (2013), 1350048 | DOI | MR | Zbl