Sang, Bo; Niu, Chuanze Solution of center-focus problem for a class of cubic systems. (English) Zbl 1343.34078 Chin. Ann. Math., Ser. B 37, No. 1, 149-160 (2016). The center-focus problem is solved for the system \[ \begin{aligned} \dot x &= y+ c_{2,0}x^2+ c_{3,0} x^3,\\ \dot y &=-x+ d_{2,0} x^2+ d_{2,1}xy+ d_{3,0}x^3+ d_{3,1} x^2y+ d_{3,2}xy^2.\end{aligned} \] Reviewer: A. P. Sadovskii (Minsk) Cited in 1 Document MSC: 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C25 Periodic solutions to ordinary differential equations Keywords:center variety; isochronous center; center conditions; integrating factor Software:Epsilon PDFBibTeX XMLCite \textit{B. Sang} and \textit{C. Niu}, Chin. Ann. Math., Ser. B 37, No. 1, 149--160 (2016; Zbl 1343.34078) Full Text: DOI References: [1] Poincaré, H., Mémoire sur les courbes définies par les équations différentielles, J. Math. Pures Appl., 1, 1885, 167-244. · JFM 17.0680.01 [2] Malkin K. E., Criteria for the center for a certain differential equation, Volz. Mat. Sb. Vyp., 2, 1964, 87-91 (in Russian). [3] Chavarriga J. and Giné J., Integrability of cubic systems with degenerate infinity, Differ. Equas Dynam. Syst., 6, 1998, 425-438. · Zbl 0998.34001 [4] Sadovskii A. P. and Shcheglova T. V., Solution of center-focus problem for a nine-parameter cubic system, Differ. Equas., 47, 2011, 208-223. · Zbl 1239.34028 [5] Hill J. M., Lloyd N. G. and Pearson J. M., Algorithmic derivation of isochronicity conditions, Nonlin. Anal.: Theory, Methods and Applications, 67, 2007, 52-69. · Zbl 1122.34022 [6] Hill J. M., Lloyd N. G. and Pearson J. M., Centres and limit cycles for an extended Kukles system, Electron. J. Differential Equations, 2007, 2007, 1-23. · Zbl 1138.34312 [7] Pearson J. M. and Lloyd N. G., Kukles revisited: Advances in computing techniques, Comput. Math. Appl., 60, 2010, 2797-2805. · Zbl 1207.68449 [8] Chen X. W. and Romanovski V. G., Linearizability conditions of time-reversible cubic systems, J. Math. Anal. Appl., 362, 2010, 438-449. · Zbl 1192.34037 [9] Zhang W. N., Hou X. R. and Zeng Z. B., Weak centers and bifurcation of critical periods in reversible cubic system, Comput. Math. Appl., 40, 2000, 771-782. · Zbl 0962.34025 [10] Chavarriga J. and Sabatini M., A survey of isochronous centers, Qualitative Theory of Dynamical Systems, 1, 1999, 1-70. [11] Chen X. W., Romanovski V. G. and Zhang W. N., Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities, Nonlin. Anal.: Theory, Methods and Applications, 69, 2008, 1525-1539. · Zbl 1159.34025 [12] Romanovski V. G., Chen X. W. and Hu Z. P., Linearizability of linear systems perturbed by fifth degree homogeneous polynomials, J. Phys. A: Math. Theor., 40, 2007, 5905-5919. · Zbl 1127.34020 [13] Boussaada I., Contribution to the study of periodic solutions and isochronous centers for planar polynomial systems of ordinary differential equations, PhD thesis, Université de Rouen, 2008, 1-94. [14] Romanovski V. G. and Shafer D. S., The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser, Boston, 2009. · Zbl 1192.34003 [15] Romanovski V. G. and Robnik M., The centre and isochronicity problems for some cubic systems, J. Phys. A: Math. Gen., 34, 2001, 10267-10292. · Zbl 1014.34028 [16] Wang D. M., Mechanical manipulation for a class of differential systems, J. Symb. Comput., 12, 1991, 233-254. · Zbl 0745.93067 [17] Zhang Z. F., Ding T. R., Huang W. Z., et al., Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, Vol. 101, American Mathematical Society, Rhode Island, 1992. · Zbl 0745.93067 [18] Giné J., On the number of algebraically independent Poincaré-Liapunov constants, Appl. Math. Comput., 188, 2007, 1870-1877. · Zbl 1124.34018 [19] Wang D. M., Elimination Practice: Software Tools and Applications, Imperial College Press, London, 2004. · Zbl 1099.13047 [20] Wang D. M., Epsilon functions. http://www-calfor.lip6.fr/wang/epsilon.pdf 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.