Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
polynomial differential system; limit cycle; averaging theory
polynomial differential system; limit cycle; averaging theory
Summary:
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.
References:
[1] Alavez-Ramírez, J., Blé, G., López-López, J., Llibre, J.: On the maximum number of limit cycles of a class of generalized Liénard differential systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 22 (2012), Article ID 1250063, 14 pages. DOI 10.1142/S0218127412500630 | MR 2916691 | Zbl 1270.34050
[2] Blows, T. R., Lloyd, N. G.: The number of small-amplitude limit cycles of Liénard equations. Math. Proc. Camb. Philos. Soc. 95 (1984), 359-366. DOI 10.1017/S0305004100061636 | MR 0735378 | Zbl 0532.34022
[3] Buică, A., Llibre, J.: Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128 (2004), 7-22. DOI 10.1016/j.bulsci.2003.09.002 | MR 2033097 | Zbl 1055.34086
[4] Chen, X., Llibre, J., Zhang, Z.: Sufficient conditions for the existence of at least $n$ or exactly $n$ limit cycles for the Liénard differential systems. J. Differ. Equations 242 (2007), 11-23. DOI 10.1016/j.jde.2007.07.004 | MR 2361100 | Zbl 1131.34026
[5] Christopher, C., Lynch, S.: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces. Nonlinearity 12 (1999), 1099-1112. DOI 10.1088/0951-7715/12/4/321 | MR 1709857 | Zbl 1074.34522
[6] Coppel, W. A.: Some quadratic systems with at most one limit cycle. Dynamics Reported A Series in Dynamical Systems and Their Applications 2. B. G. Teubner, Stuttgart; John Wiley & Sons, Chichester (1989), 61-88 U. Kirchgraber et al. DOI 10.1007/978-3-322-96657-5_3 | MR 1000976 | Zbl 0674.34026
[7] García, B., Llibre, J., Río, J. S. Peréz del: Limit cycles of generalized Liénard polynomial differential systems via averaging theory. Chaos Solitons Fractals 62-63 (2014), 1-9. DOI 10.1016/j.chaos.2014.02.008 | MR 3200747 | Zbl 1348.34066
[8] Gradshteyn, I. S., Ryzhik, I. M.: Table of Integrals, Series, and Products. Academic Press, Amsterdam (2007). DOI 10.1016/C2009-0-22516-5 | MR 2360010 | Zbl 1208.65001
[9] Han, M., Yu, P.: Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles. Applied Mathematical Sciences 181. Springer, Berlin (2012). DOI 10.1007/978-1-4471-2918-9 | MR 2918519 | Zbl 1252.37002
[10] Li, J.: Hilbert's 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurcation Chaos Appl. Sci. Eng. 13 (2003), 47-106. DOI 10.1142/S0218127403006352 | MR 1965270 | Zbl 1063.34026
[11] Llibre, J., Makhlouf, A.: Limit cycles of a class of generalized Liénard polynomial equations. J. Dyn. Control Syst. 21 (2015), 189-192. DOI 10.1007/s10883-014-9253-4 | MR 3314541 | Zbl 1325.34042
[12] Llibre, J., Mereu, A. C., Teixeira, M. A.: Limit cycles of the generalized polynomial Liénard differential equations. Math. Proc. Camb. Philos. Soc. 148 (2010), 363-383. DOI 10.1017/S0305004109990193 | MR 2600146 | Zbl 1198.34051
[13] Llibre, J., Valls, C.: On the number of limit cycles of a class of polynomial differential systems. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 468 (2012), 2347-2360. DOI 10.1098/rspa.2011.0741 | MR 2949385 | Zbl 1371.34044
[14] Llibre, J., Valls, C.: Limit cycles for a generalization of polynomial Liénard differential systems. Chaos Solitons Fractals 46 (2013), 65-74. DOI 10.1016/j.chaos.2012.11.010 | MR 3011852 | Zbl 1258.34060
[15] Llibre, J., Valls, C.: On the number of limit cycles for a generalization of Liénard polynomial differential systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 23 (2013), Article ID 1350048, 16 pages. DOI 10.1142/S021812741350048X | MR 3047963 | Zbl 1270.34052
Search
Partner of
