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Yu, Xiaolan; Zhang, Xiang

The hyperelliptic limit cycles of the Liénard systems. (English) Zbl 1208.34038

J. Math. Anal. Appl. 376, No. 2, 535-539 (2011).
Consider the Liénard system \[ \dot x= y,\quad \dot y= -f_m(x) y- g_n(x),\tag{\(*\)} \] where \(f_m\) and \(g_n\) are real polynomials of degree \(m\) and \(n\), respectively. A planar algebraic curve \(F(x,y)= 0\) is called hyperelliptic, if \(F\) can be written in the form \[ F(x,y)\equiv (y+ p(x))^2- q(x), \] where \(p\) and \(q\) are real polynomials. The authors prove the result: There exist Liénard systems \((*)\) with \(m= 5\), \(n= 7,8,9\) which have hyperelliptic limit cycles.
Reviewer: Klaus R. Schneider (Berlin)

Cited in 1 Review
Cited in 10 Documents

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Keywords:

Liénard system; algebraic limit cycle
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References:

[1] Cao, Jinlong; Jiang, Haixia, Planar polynomial vector fields having first integrals and algebraic limit cycles, J. Math. Anal. Appl., 361, 177-186 (2010) · Zbl 1191.34041
[2] Chavarriga, J.; García, I. A.; Sorolla, J., Non-nested configuration of algebraic limit cycles in quadratic systems, J. Differential Equations, 225, 513-527 (2006) · Zbl 1107.34026
[3] Chavarriga, J.; Giacomini, H.; Grau, M., Quadratic systems with an algebraic limit cycle of degree 2 or 4 do not have a Liouvillian first integral, (EQUADIFF 2003 (2005), World Sci. Publ.: World Sci. Publ. Hackensack, NJ), 325-327 · Zbl 1106.34311
[4] Chavarriga, J.; García, I. A.; Llibre, J.; Zoladek, H., Invariant algebraic curves for the cubic Liénard system with linear damping, Bull. Sci. Math., 130, 428-441 (2006) · Zbl 1123.34024
[5] Chavarriga, J.; Giacomini, H.; Llibre, J., Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. Appl., 261, 85-99 (2001) · Zbl 0995.34024
[6] Ciambellotti, L., Uniqueness of limit cycles for Liénard systems. A generalization of Massera’s theorem, Qual. Theory Dyn. Syst., 7, 405-410 (2009) · Zbl 1329.34063
[7] Al-Dosary, K. I.T., Non-algebraic limit cycles for parametrized planar polynomial systems, Internat. J. Math., 18, 179-189 (2007) · Zbl 1121.34036
[8] Dumortier, F.; Panazzolo, D.; Roussarie, R., More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc., 135, 1895-1904 (2007) · Zbl 1130.34018
[9] García, I. A.; Giné, J.; Llibre, J., Liénard and Riccati differential equations related via Lie algebras, Discrete Contin. Dyn. Syst. Ser. B, 10, 485-494 (2008) · Zbl 1162.34025
[10] Gasull, A.; Giacomini, H.; Torregrosa, J., Explicit non-algebraic limit cycles for polynomial systems, J. Comput. Appl. Math., 200, 448-457 (2007) · Zbl 1171.34021
[11] Giné, J.; Grau, M., Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations, Nonlinearity, 19, 1939-1950 (2006) · Zbl 1114.34029
[12] Giné, J.; Grau, M., Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations, Nonlinearity, 19, 1939-1950 (2006) · Zbl 1114.34029
[13] Liénard, A., Étude des oscillations entretenues, Rev. Gen. Elect., 23, 946-954 (1928)
[14] Neto, A. Lins; de Melo, W.; Pugh, C. C., On Liénard equations, (Proc. Symp. Geom. and Topol.. Proc. Symp. Geom. and Topol., Lecture Notes in Math., vol. 597 (1977), Springer-Verlag), 335-357 · Zbl 0362.34022
[15] Llibre, J., A survey on the limit cycles of the generalized polynomial Liénard differential equations, (Mathematical Models in Engineering, Biology and Medicine, Proceedings of the International Conference on Boundary Value Problems. Mathematical Models in Engineering, Biology and Medicine, Proceedings of the International Conference on Boundary Value Problems, Santiago de Compostela, September 16-19, 2008. Mathematical Models in Engineering, Biology and Medicine, Proceedings of the International Conference on Boundary Value Problems. Mathematical Models in Engineering, Biology and Medicine, Proceedings of the International Conference on Boundary Value Problems, Santiago de Compostela, September 16-19, 2008, AIP Conf. Proc., vol. 1124 (2009)), 224-233
[16] Llibre, J., Open problems on the algebraic limit cycles of planar polynomial vector fields, Bul. Acad. Stiinte Repub. Mold. Mat., 1, 19-26 (2008) · Zbl 1280.34035
[17] Llibre, J.; Ramírez, R.; Sadovskaia, N., On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations, 248, 1401-1409 (2010) · Zbl 1204.34038
[18] J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves, J. Differential Equations, doi:10.1016/j.jde.2010.06.009; J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves, J. Differential Equations, doi:10.1016/j.jde.2010.06.009 · Zbl 1269.34038
[19] Llibre, J.; Swirszcz, G., Classification of quadratic systems admitting the existence of an algebraic limit cycle, Bull. Sci. Math., 131, 405-421 (2007) · Zbl 1132.34028
[20] Llibre, J.; Swirszcz, G., Relationships between limit cycles and algebraic invariant curves for quadratic systems, J. Differential Equations, 229, 529-537 (2006) · Zbl 1109.34025
[21] Llibre, J.; Mereu, A. C.; Teixeira, M. A., Limit cycles of the generalized polynomial Liénard differential equations, Math. Proc. Cambridge Philos. Soc., 148, 363-383 (2010) · Zbl 1198.34051
[22] Llibre, J.; Zhang, Xiang, On the algebraic limit cycles of Liénard systems, Nonlinearity, 21, 2011-2022 (2008) · Zbl 1158.34021
[23] Llibre, J.; Zhao, Yulin, Algebraic limit cycles in polynomial systems of differential equations, J. Phys. A, 40, 14207-14222 (2007) · Zbl 1135.34025
[24] Odani, K., On the algebraic limit cycles of Liénard systems, J. Differential Equations, 115, 146-152 (1995) · Zbl 0816.34023
[25] Sáez, E.; Szántó, I., Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems, Period. Math. Hungar., 56, 137-142 (2008) · Zbl 1164.34379
[26] Wilson, J. C., Algebraic periodic solutions of Liénard equations, Contrib. Diff. Eqns., 3, 1-20 (1964) · Zbl 0142.06505
[27] Ye, Yanqian, Theory of Limit Cycles, Transl. Math. Monogr., vol. 66 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0653.34023
[28] Ye, Yanqian, Qualitative Theory of Polynomial Differential Systems (1995), Shanghai Scientific & Technical Publishers: Shanghai Scientific & Technical Publishers Shanghai, (in Chinese) · Zbl 0854.34003
[29] Xiaolan Yu, Algebraic limit cycles of Liénard systems, Thesis, Shanghai Jiaotong University, 2010.; Xiaolan Yu, Algebraic limit cycles of Liénard systems, Thesis, Shanghai Jiaotong University, 2010.
[30] Xiang Zhang, The 16th Hilbert problem on algebraic limit cycles, preprint, 2010.; Xiang Zhang, The 16th Hilbert problem on algebraic limit cycles, preprint, 2010.
[31] Zhang, Zhifen; Ding, Tongren; Huang, Wenzao; Dong, Zhenxi, Qualitative Theory of Differential Equations, Transl. Math. Monogr., vol. 101 (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0779.34001
[32] Zoladek, H., Algebraic invariant curves for the Liénard equation, Trans. Amer. Math. Soc., 350, 1681-1701 (1998) · Zbl 0895.34026
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