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A new method to determine isochronous center conditions for polynomial differential systems. (English) Zbl 1034.34032

The authors give a new algorithm to compute period constants of complex polynomial systems, and the set of period constants of real polynomial systems is a spacial case of it. The algorithm is recursive and avoids complex integrating operations and solving equations. As an example of the algorithm, center conditions and isochronous center conditions for a class of higher degree systems have been discussed.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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[1] Amelbkin, B. B.; Lukasevnky, H. A.; Catovcki, A. N., Nonlinear Vibration (1982), VGU Lenin Publications, (in Russian)
[2] Blows, T. R.; Rousseau, C., Bifurcation at infinity in polynomial vector fields, J. Differential Equations, 104, 215-242 (1993) · Zbl 0778.34024
[3] Cairó, L.; Chavrriga, J.; Giné, J.; Llibre, J., A class of reversible cubic systems with an isochronous center, Comput. Math. Appl., 38, 39-53 (1999) · Zbl 0982.34024
[4] Chavarriga, J.; Giné, J.; García, I., Isochronous centers of cubic systems with degenerate infinity, Differential Equations Dynam. Syst., 7, 2, 221-238 (1999) · Zbl 0982.34025
[5] Chavarriga, J.; Giné, J.; García, I., Isochronous centers of a linear center perturbed by fourth degree homogrneous polynomial, Bull. Sci. Math., 123, 77-96 (1999) · Zbl 0921.34032
[6] Chavarriga, J.; Giné, J.; García, I., Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomial, J. Comput. Appl. Math., 126, 351-368 (2000) · Zbl 0978.34028
[7] Chavarriga, J.; Giné, J.; García, I., Isochronicity into a family of time-reversible cubic vector fields, Appl. Math. Comput., 121, 129-145 (2001) · Zbl 1025.37011
[8] Christopher, C. J.; Devlin, J., Isochronous centers in planar polynomial systems, SIAM J. Math. Anal., 28, 162-177 (1997) · Zbl 0881.34057
[9] Farr, W. W.; Li, C.; Labouriau, I. S.; Langford, W. F., Degenerate Hopf-bifurcation formulas and Hilbert’s 16th problem, SIAM J. Math. Anal., 20, 13-29 (1989) · Zbl 0682.58035
[10] Gasull, A.; Manosa, V., An explicit expression of the first Liapunov and period constants with applications, J. Math. Anal. Appl., 211, 190-212 (1997) · Zbl 0882.34040
[11] Gine, J., Conditions for the existence of a center for the Kukless homogeneous systems, Comput. Math. Appl., 43, 1261-1269 (2002) · Zbl 1012.34025
[12] Hassard, B.; Wan, Y. H., Bifurcation formulae derived from center manifold theroy, J. Math. Anal. Appl., 63, 297-312 (1978) · Zbl 0435.34034
[13] Linyiping; Lijibin, Normal form and critical points of the period of closed orbits for planar autonomous systems, Acta Math. Sinica, 34, 490-501 (1991), (in Chinese)
[14] Liu, Y.; Chen, H., Formulas of singurlar 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
[15] Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential system, Science in China (Series A), 33, 10-24 (1990)
[16] Lloyd, N. G.; Christopher, J.; Devlin, J.; Pearson, J. M.; Uasmin, N., Quadratic like cubic systems, Differential Equations Dynam. Syst., 5, 3-4, 329-345 (1997) · Zbl 0898.34026
[17] Lloyd, N. G.; Pearson, J. M., Symmetry in planar dynamical systems, J. Symbolic Comput., 33, 357-366 (2002) · Zbl 1003.34026
[18] Loud, W. S., Behavior of the period of solutions of certain plane autonomous systems near centers, Contrib. Differential Equations, 3, 21-36 (1964) · Zbl 0139.04301
[19] Mardesic, P.; Rousseau, C.; Toni, B., Linearzation of isochronous centers, J. Differential Equations, 121, 67-108 (1995) · Zbl 0830.34023
[20] Pleshkan, I., A new method of investigating the isochronicity of a system of two differential equations, Differential Equations, 5, 796-802 (1969) · Zbl 0252.34034
[21] Romanovski, V. G.; Suba, A., Centers of some cubic systems, Ann. Differential Equations, 17, 363-370 (2001) · Zbl 1014.34019
[22] Salih, N.; Pons, R., Center conditions for a lopsided quartic polynomial vector field, Bull. Sci. Math., 126, 369-378 (2002) · Zbl 1015.34016
[23] Ye, Y., Qualitative Theory of Polynomial Differential Systems (1995), Shanghai Sci. Tech. Publications: Shanghai Sci. Tech. Publications Shanghai, (in Chinese)
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