On the number of hyperelliptic limit cycles of Liénard systems. (English) Zbl 1453.34045
This paper is devoted to a study of the maximum number \(H(m, n)\) of hyperelliptic limit cycles for the Liénard system
\[\dot{x}=y, \quad \dot{y}=-f_m(x)y-g_n(x),\]
where \(f_m(x)\) and \(g_n(x)\) are real polynomials of degree \(m\) and \(n\) respectively.
A limit cycle is called an algebraic limit cycle if it is contained in an invariant algebraic curve defined by equation \(F(x, y) = 0\). If \(F(x, y)\) takes the form \(F(x, y) = (y + P(x))^2 - Q(x)\), where \(P\) and \(Q\) are polynomials, then the invariant curve is called as hyperelliptic. Correspondingly, a limit cycle is said to be hyperelliptic if it is contained in an invariant hyperelliptic curve.
The main results of the paper represent the upper as well as the lower bounds for \(H(m,n)\). The authors claim that in most cases these bounds are sharp.
A limit cycle is called an algebraic limit cycle if it is contained in an invariant algebraic curve defined by equation \(F(x, y) = 0\). If \(F(x, y)\) takes the form \(F(x, y) = (y + P(x))^2 - Q(x)\), where \(P\) and \(Q\) are polynomials, then the invariant curve is called as hyperelliptic. Correspondingly, a limit cycle is said to be hyperelliptic if it is contained in an invariant hyperelliptic curve.
The main results of the paper represent the upper as well as the lower bounds for \(H(m,n)\). The authors claim that in most cases these bounds are sharp.
Reviewer: Alexander Grin (Grodno)
MSC:
34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
34A34 | Nonlinear ordinary differential equations and systems |
References:
[1] | Llibre, J., A survey on the limit cycles of the generalization polynomial Liénard differential equations, AIP Conf. Proc., 1124, 224-233 (2009) · doi:10.1063/1.3142937 |
[2] | Dumortier, F.; Li, C., On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations, Nonlinearity, 9, 1489-1500 (1996) · Zbl 0907.58056 · doi:10.1088/0951-7715/9/6/006 |
[3] | Odani, K., The limit cycle of the van der Pol equation is not algebraic, J. Differ. Equ., 115, 146-82 (1995) · Zbl 0816.34023 · doi:10.1006/jdeq.1995.1008 |
[4] | Chavarriga, J.; Garcia, Ia; 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 · doi:10.1016/j.bulsci.2006.03.013 |
[5] | Zoladek, H., Algebraic invariant curves for the Liénard equation, Trans. Am. Math. Soc., 350, 1681-1701 (1998) · Zbl 0895.34026 · doi:10.1090/S0002-9947-98-02002-9 |
[6] | Llibre, J.; Zhang, X., On the algebraic limit cycles of Liénard systems, Nonlinearity, 21, 2011-2022 (2008) · Zbl 1158.34021 · doi:10.1088/0951-7715/21/9/004 |
[7] | Yu, X.; Zhang, X., The hyperelliptic limit cycles of the Liénard systems, J. Math. Anal. Appl., 376, 535-539 (2011) · Zbl 1208.34038 · doi:10.1016/j.jmaa.2010.12.015 |
[8] | Liu, C.; Chen, G.; Yang, J., On the hyperelliptic limit cycles of Liénard systems, Nonlinearity, 25, 1601-1611 (2012) · Zbl 1366.34045 · doi:10.1088/0951-7715/25/6/1601 |
[9] | Yang, L.; Hou, X.; Zeng, Z., A complete discrimination system for polynomials, Sci. China Ser. E, 39, 628-646 (1996) · Zbl 0866.68104 |
[10] | Yang, L.; Zhang, J.; Hou, X., Nonlinear Algebraic Equation System and Automated Theorem Proving (1996), Shanghai: Shanghai Scientific and Technological Education Publishing House, Shanghai |
[11] | Jouanolou, Jp, Equations de Pfaff algébriques, Lectures Notes in Mathematics (1979), Berlin: Springer, Berlin · Zbl 0477.58002 |
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