×

New lower bound for the number of critical periods for planar polynomial systems. (English) Zbl 1462.34067

In this paper, the author constructs two classes of planar polynomial Hamiltonian systems with a center at the origin and obtain the lower bounds for the number of critical periods for derived classes. The main result of the paper can be described as follows: there exists a polynomial potential systems of degree \(n\) whose number of critical periods is at least \(n-2\), and there exists a polynomial systems of degree \(n\) whose number of critical periods is at least \(n^2/2+n-5/2\) for odd \(n\) and \(n^2/2-2\) for even \(n\). The idea used in this paper is essentially different from the methods in other papers.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to 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
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Chavarriga, J.; Sabatini, M., A survey of isochronous centers, Qual. Theory Dyn. Syst., 1, 1-70 (1999)
[2] Chen, X.; Romanovski, V. G.; Zhang, W., Critical periods of perturbations of reversible rigidly isochronous centers, J. Differ. Equ., 251, 6, 1505-1525 (2011) · Zbl 1237.34078
[3] Chicone, C.; Jacobs, M., Bifurcation of critical periods for plane vector fields, Trans. Am. Math. Soc., 312, 2, 433-486 (1989) · Zbl 0678.58027
[4] Chow, S.-N.; Sanders, J., On the number of critical points of the period, J. Differ. Equ., 64, 1, 51-66 (1986) · Zbl 0594.34028
[5] Cima, A.; Gasull, A.; Silva, P. R., On the number of critical periods for planar polynomial systems, Nonlinear Anal., 69, 1889-1903 (2008) · Zbl 1157.34021
[6] Ferčec, B.; Levandovskyy, V.; Romanovski, V. G.; Shafer, D. S., Bifurcation of critical periods of polynomial systems, J. Differ. Equ., 259, 8, 3825-3853 (2015) · Zbl 1337.34032
[7] Gasull, A.; Liu, C.; Yang, J., On the number of critical periods for planar polynomial systems of arbitrary degree, J. Differ. Equ., 249, 3, 684-692 (2010) · Zbl 1203.34049
[8] Gasull, A.; Yu, J., On the critical periods of perturbed isochronous centers, J. Differ. Equ., 244, 3, 696-715 (2008) · Zbl 1143.34024
[9] Gasull, A.; Zhao, Y., Bifurcation of critical periods from the rigid quadratic isochronous vector field, Bull. Sci. Math., 132, 4, 292-312 (2008) · Zbl 1160.34035
[10] Gavrilov, L., Remark on the number of critical points of the period, J. Differ. Equ., 101, 1, 58-65 (1993) · Zbl 0765.34030
[11] Gavrilov, L.; Giné, J.; Grau, M., On the cyclicity of weight-homogeneous centers, J. Differ. Equ., 246, 3126-3135 (2009) · Zbl 1182.34045
[12] Huang, W.; Basov, V.; Han, M.; Romanovski, V. G., Bifurcation of critical periods of a quartic system, Electron. J. Qual. Theory Differ. Equ., Article 76 pp. (2018) · Zbl 1413.34147
[13] Il’yashenko, Yu. S., The appearance of limit cycles under perturbations of the equation \(d w / d z = - R z / R w\) where \(R(z, w)\) is a polynomial, Math. Sb. (N.S.), 78, 120, 360-373 (1969) · Zbl 0183.36501
[14] Li, C.; Lu, K., The period function of hyperelliptic Hamiltonians of degree 5 with real critical points, Nonlinearity, 21, 465-483 (2008) · Zbl 1142.34016
[15] Li, J.; Chan, H. S.Y.; Chung, K. W., Some lower bounds for \(H(n)\) in Hilbert’s 16th problem, Qual. Theory Dyn. Syst., 3, 345-360 (2003) · Zbl 1050.34039
[16] Li, W.; Llibre, J.; Yang, J.; Zhang, Z., Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dyn. Differ. Equ., 21, 133-152 (2009) · Zbl 1176.34038
[17] Liang, H.; Zhao, Y., On the period function of reversible quadratic centers with their orbits inside quartics, Nonlinear Anal., 71, 11, 5655-5671 (2009) · Zbl 1182.34059
[18] Liang, H.; Zhao, Y., On the period function of a class of reversible quadratic centers, Acta Math. Sin. Engl. Ser., 27, 5, 905-918 (2011) · Zbl 1226.34034
[19] De Maesschalck, P.; Dumortier, F., The period function of classical Liénard equations, J. Differ. Equ., 233, 380-403 (2007) · Zbl 1121.34046
[20] Mañosas, F.; Rojas, D.; Villadelprat, J., Analytic tools to bound the criticality at the outer boundary of the period annulus, J. Dyn. Differ. Equ., 30, 883-909 (2018) · Zbl 1444.34055
[21] Mañosas, F.; Villadelprat, J., Criteria to bound the number of critical periods, J. Differ. Equ., 246, 6, 2415-2433 (2009) · Zbl 1171.34022
[22] Romanovski, V. G.; Han, M.; Huang, W., Bifurcation of critical periods of a quintic system, Electron. J. Differ. Equ., Article 66 pp. (2018) · Zbl 1391.34075
[23] Yang, L.; Zeng, X., The period function of potential systems of polynomials with real zeros, Bull. Sci. Math., 133, 555-577 (2009) · Zbl 1191.34055
[24] Zhao, Y., The monotonicity of period function for codimension four quadratic system \(Q_4\), J. Differ. Equ., 185, 1, 370-387 (2002) · Zbl 1047.34024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.