Abstract
We describe an approach to studying the center problem and local bifurcations of critical periods at infinity for a class of differential systems. We then solve the problem and investigate the bifurcations for a class of rational differential systems with a cubic polynomial as its numerator.
Similar content being viewed by others
References
Bautin, N. On the number of limit cyckes which appear with the variation fo coefficients from an equilibrium position of focus or center type. Amer. Math. Soc. Trans., 100: 397–413 (1954)
Blows, T.R., Rousseau, C. Bifurcation at infinity in polynomial vector fields. Journal of Differential Equations, 104: 215–242 (1993)
Chen, X., Zhang, W. Decomposition of algebraic sets and applications to weak centers of cubic systems. J. Comput. Appl. Math., 232: 565–581 (2009)
Chicone, C., Jacobs, M. Bifurcation of critical periods for plane vector fields. Transactions Amer. Math. Soc., 312: 319–329 (1989)
Cima, A., Gasull, A., da Silvab, P.R. On the number of critical periods for planar polynomial systems. Nonlinear Analysis, 69: 1889–1903 (2008)
Decker, W., Pfister, G., Schönemann, H. A Singular 2.0 library for computing the primary decomposition and radical of ideals. primdec.lib, http://www.singular.uni-kl.de, 2001
Du, Z. On the critical periods of Liénard systems with cubic restoring forces. International Journal of Mathematics and Mathematical Sciences, 61: 3259–3274 (2004)
Gasull, A., Zhao, Y. Bifurcation of critical periods from the rigid quadratic isochronous vector field. Bulletin des Sciences Mathematiques, 132: 291–312 (2008)
Gianni, P., Trager, B., Zacharias, G. Gröbner bases and primary decomposition of polynomials. J. Symbolic Comput., 6: 146–167 (1988)
Greuel, G.M., Pfister, G., Schönemann, H. Singular 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, http://www.singular.uni-kl.de.., 2005
Jarrah, A., Laubenbacher, R., Romanovski, V. The Sibirsky component of the center variety of polynomial systems. Journal of Symbolic Computation, 35: 577–589 (2003)
Lin, Y., Li, J. The canonical form of the autonomous planar system and the critical point of the closed orbit period. Acta Mathematica Sinica, 34: 490–501 (1991)
Liu, Y., Chen, H. Formulas of singular point quantities and the first 10 saddle quantities for a class of cubic system. Acta Mathematicae Applicatae sinica, 25: 295–302 (2002) (in Chinese)
Liu, Y., Huang, W. Center and isochronous center at infinity for differential systems, Bulletin des Sciences. Mathématiques, 128: 77–89 (2004)
Liu, Y., Huang, W. A new method to determine isochronous center conditions for polynomial differential systems. Bulletin des Sciences Mathématiques, 127: 133–148 (2003)
Liu, Y., Li, J. Periodic constants and time-angle difference of isochronous centers for complex analytic systems. Int. J. Bifurcation and Chaos, 16: 3747–3757 (2006)
Liu, Y., Li, J., Huang, W. Singular point values, center problem and bifurcations of limit cycles of two dimensional differential autonomous systems. Beijing: Science Press, 2008
Romanovski, V.G., Han, M. Critical period bifurcations of a cubic system. J. Phys. A: Math. and Gen., 36: 5011–5022 (2003)
Romanovski V.G., Shafer, D.S. The center and cyclicity problems: a computational algebra approach. Boston: Birkhäuser Boston, Inc., MA, 2009
Rousseau, C., Toni, B. Local bifurcations of critical periods in the reduced Kukles system. Can. J. Math., 49: 338–358 (1997)
Rousseau, C., Toni, B. Local bifurcations of critical periods in vector fields with homogeneous nonliearities of the third degree. Can. J. Math., 36: 473–484 (1993)
Wang, D. Elimination methods. Texts and Monographs in Symbolic Computation. Springer-Verlag, Vienna, 2001
Yu, P., Han, M. Critical periods of planar revetible vector field with third-degree polynomial functions. International Journal of Bifurcation and Chaos, 19(1): 419–433 (2009)
Zhang, W., Hou, X., Zeng, Z., Weak centres and bifurcation of critical periods in reversible cubic systems. Comput. Math. Appl., 40(6–7): 771–782 (2000)
Zou, L., Chen, X., Zhang, W. Local bifurcations of critical periods for cubic Liénard equations with cubic damping. Journal of Computational and Applied Mathematics, 222: 404–410 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is supported by the National Natural Science Foundation of China (10961011) and the Slovene Human Resources and Scholarship Fund. The second author acknowledges support of this work by the Slovenian Research Agency, by the Nova Kreditna Banka Maribor, by TELEKOM Slovenije and by the Transnational Access Programme at RISC-Linz of the European Commission Framework 6 Programme for Integrated Infrastructures Initiatives under the project SCIEnce (Contract No. 026133).
Rights and permissions
About this article
Cite this article
Huang, Wt., Romanovski, V.G. & Zhang, WN. Weak centers and local bifurcations of critical periods at infinity for a class of rational systems. Acta Math. Appl. Sin. Engl. Ser. 29, 377–390 (2013). https://doi.org/10.1007/s10255-013-0220-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-013-0220-8