Weak centers and local bifurcations of critical periods at infinity for a class of rational systems.(English)Zbl 1329.34065
Summary: 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.
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
Cite
Full Text:DOI
References:
| [1] | 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) |
| [2] | Blows, T.R., Rousseau, C. Bifurcation at infinity in polynomial vector fields. Journal of Differential Equations, 104: 215–242 (1993) ·Zbl 0778.34024 ·doi:10.1006/jdeq.1993.1070 |
| [3] | Chen, X., Zhang, W. Decomposition of algebraic sets and applications to weak centers of cubic systems. J. Comput. Appl. Math., 232: 565–581 (2009) ·Zbl 1178.13015 ·doi:10.1016/j.cam.2009.06.029 |
| [4] | Chicone, C., Jacobs, M. Bifurcation of critical periods for plane vector fields. Transactions Amer. Math. Soc., 312: 319–329 (1989) ·Zbl 0678.58027 ·doi:10.1090/S0002-9947-1989-0930075-2 |
| [5] | Cima, A., Gasull, A., da Silvab, P.R. On the number of critical periods for planar polynomial systems. Nonlinear Analysis, 69: 1889–1903 (2008) ·Zbl 1157.34021 ·doi:10.1016/j.na.2007.07.031 |
| [6] | 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 |
| [7] | Du, Z. On the critical periods of Liénard systems with cubic restoring forces. International Journal of Mathematics and Mathematical Sciences, 61: 3259–3274 (2004) ·Zbl 1126.34328 ·doi:10.1155/S0161171204402245 |
| [8] | Gasull, A., Zhao, Y. Bifurcation of critical periods from the rigid quadratic isochronous vector field. Bulletin des Sciences Mathematiques, 132: 291–312 (2008) ·Zbl 1160.34035 |
| [9] | Gianni, P., Trager, B., Zacharias, G. Gröbner bases and primary decomposition of polynomials. J. Symbolic Comput., 6: 146–167 (1988) ·Zbl 0667.13008 ·doi:10.1016/S0747-7171(88)80040-3 |
| [10] | 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 |
| [11] | Jarrah, A., Laubenbacher, R., Romanovski, V. The Sibirsky component of the center variety of polynomial systems. Journal of Symbolic Computation, 35: 577–589 (2003) ·Zbl 1035.34017 ·doi:10.1016/S0747-7171(03)00016-6 |
| [12] | 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) ·Zbl 0744.34041 |
| [13] | 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) ·Zbl 1014.34021 |
| [14] | Liu, Y., Huang, W. Center and isochronous center at infinity for differential systems, Bulletin des Sciences. Mathématiques, 128: 77–89 (2004) ·Zbl 1048.34066 ·doi:10.1016/j.bulsci.2003.07.003 |
| [15] | Liu, Y., Huang, W. A new method to determine isochronous center conditions for polynomial differential systems. Bulletin des Sciences Mathématiques, 127: 133–148 (2003) ·Zbl 1034.34032 ·doi:10.1016/S0007-4497(02)00006-4 |
| [16] | 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) ·Zbl 1140.34301 ·doi:10.1142/S0218127406017142 |
| [17] | 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 |
| [18] | Romanovski, V.G., Han, M. Critical period bifurcations of a cubic system. J. Phys. A: Math. and Gen., 36: 5011–5022 (2003) ·Zbl 1037.34034 ·doi:10.1088/0305-4470/36/18/306 |
| [19] | Romanovski V.G., Shafer, D.S. The center and cyclicity problems: a computational algebra approach. Boston: Birkhäuser Boston, Inc., MA, 2009 ·Zbl 1192.34003 |
| [20] | Rousseau, C., Toni, B. Local bifurcations of critical periods in the reduced Kukles system. Can. J. Math., 49: 338–358 (1997) ·Zbl 0885.34033 ·doi:10.4153/CJM-1997-017-4 |
| [21] | 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) ·Zbl 0792.58030 ·doi:10.4153/CMB-1993-063-7 |
| [22] | Wang, D. Elimination methods. Texts and Monographs in Symbolic Computation. Springer-Verlag, Vienna, 2001 |
| [23] | 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) ·Zbl 1170.34316 ·doi:10.1142/S0218127409022981 |
| [24] | 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) ·Zbl 0962.34025 ·doi:10.1016/S0898-1221(00)00195-4 |
| [25] | 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) ·Zbl 1163.34349 ·doi:10.1016/j.cam.2007.11.005 |
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.
