Abstract
This paper is concerned with the bi-isochronous centers problem for a cubic systems in \({Z}_2\)-equivariant vector field. Being based on bi-centers condition, we compute the period constants and obtain the periodic constants basis for each center condition separately of eleven conditions by symbolic computation and numerical analysis with help of the computer algebra system Mathematica. Moreover, we give the sufficient and necessary condition that investigated cubic system has a pair of isochronous centers. In terms of the result of simultaneous isochronous centers including bi-isochronous centers, it is hardly seen in published references, our result in this paper is new.
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Liu, C., Han, M.: Bifurcation of critical periods from the reversible rigidly isochronous centers. Nonlinear Anal. Theory Methods Appl. 95, 388–403 (2014)
Chen, X., Romanovski, V.G., Zhang, W.: Critical periods of perturbations of reversible rigidly isochronous centers. J. Differ. Equ. 251, 1505–1525 (2011)
Dolicanin-Dekic, D.: Strongly isochronous centers of cubic systems with degenerate infinity. Differ. Equ. 50(7), 971–975 (2014)
Giné, J., Llibre, J., Valls, C.: Centers and isochronous centers for generalized quintic systems. J. Comput. Appl. Math. 279, 173–186 (2014)
Zhang, W., Hou, X., Zeng, Z.: Weak center and bifurcation of critical periods in reversible cubic systems. Comput. Math. Appl. 40(6–7), 771–782 (2000)
Du, C., Liu, Y., Liu, C.: Weak center problem and bifurcation of critical periods for a Z2-equivariant cubic system. Adv. Differ. Equ. (2013). doi:10.1186/1687-1847-2013-197
Pleshkan, I.: A new method of investigating the isochronicity of system of two differential equations. Differ. Equ. 5, 796–802 (1969)
Liu, Y., Huang, W.: A new method to determine isochronous center conditions for polynomial differential systems. Bull. Sci. Math. 127, 133–148 (2003)
Du, C., Liu, Y., Mi, H.: A class of ninth degree system with four isochronous centers. Comput. Math. Appl. 56, 2609–2620 (2008)
Chen, T., Huang, W., Ren, D.: Weak centers and local critical periods for a Z2-equivariant cubic system. Nonlinear Dyn. 78, 2319–2329 (2014)
Itikawa, J., Llibre, J.: Phase portraits of uniform isochronous quartic centers. J. Comput. Appl. Math. 287, 98–114 (2015)
Li, F., Yu, P., Tian, Y., Liu, Y.: Center and isochronous center conditions for switching systems associated with elementary singular points. Commun. Nonlinear Sci. Numer. Simul. 28, 81–97 (2015)
Li, J., Liu, Y.: New results on the study of Zq-equivariant planar polynomial vector fields. Qual. Theory Dyn. Syst. 9, 167–219 (2010)
Liu, Y., Li, J.: Z2-equivariant cubic system which yields 13 limit cycles. Acta Math. Appl. Sin. 30(3), 781–800 (2014)
Li, C., Liu, C., Yang, J.: A cubic system with thirteen limit cycles. J. Differ. Equ. 246, 3609–3619 (2009)
Du, C., Huang, W., Zhang, Q.: Center problem and bifurcation of limit cycles for a cubic polynomial system. Appl. Math. Model. 39, 5200–5215 (2015)
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This research is partially supported by the National Natural Science Foundation of China (11371373, 11261013) and the Research Fund of Hunan provincial education department (14A132).
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Du, C., Liu, Y. Isochronicity for a \(\varvec{Z_{2}}\)-equivariant cubic system. Nonlinear Dyn 87, 1235–1252 (2017). https://doi.org/10.1007/s11071-016-3112-7
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DOI: https://doi.org/10.1007/s11071-016-3112-7