The problem of bicenter and isochronicity for a class of quasi symmetric planar systems. (English) Zbl 1470.34084

Summary: We study a class of quasi symmetric seventh degree systems and obtain the conditions that its two singular points can be two centers at the same step by careful computing and strict proof. In addition, the condition of an isochronous center is also given. In terms of quasi symmetric systems, our work is interesting and obtained conclusions about bicenters are new.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
Full Text: DOI


[1] Lin, Y.; Li, J., Normal form and critical points values of the period of closed orbits for planar autonomous systems, Acta Mathematica Sinica, 34, 4, 490-501 (1991) · Zbl 0744.34041
[2] Romanovskii, V. G., Calculation of Lyapunov numbers in the case of two pure imaginary roots, Differential Equations, 29, 5, 782-784 (1993) · Zbl 0833.34028
[3] Huang, W.; Liu, Y.; Zhang, W., Conditions of infinity to be an isochronous centre for a rational differential system, Mathematical and Computer Modelling, 46, 5-6, 583-594 (2007) · Zbl 1137.34015 · doi:10.1016/j.mcm.2006.11.022
[4] Liu, Y.; Huang, W., Center and isochronous center at infinity for differential systems, Bulletin des Sciences Mathematiques, 128, 2, 77-89 (2004) · Zbl 1048.34066 · doi:10.1016/j.bulsci.2003.07.003
[5] Feng, L.; Yirong, L., Classification of the centers and isochronicity for a class of quartic polynomial differential systems, Communications in Nonlinear Science and Numerical Simulation, 17, 6, 2270-2291 (2012) · Zbl 1250.34024 · doi:10.1016/j.cnsns.2011.09.027
[6] Li, F.; Qiu, J.; Li, J., Bifurcation of limit cycles, classification of centers and isochronicity for a class of non-analytic quintic systems, Nonlinear Dynamics, 76, 1, 183-197 (2014) · Zbl 1319.37035 · doi:10.1007/s11071-013-1120-4
[7] Romanovski, V. G.; Shafer, D. S., The Center and Cyclicity Problems. A Computational Algebra Approach (2009), Berlin, Germany: Birkhäuser, Berlin, Germany · Zbl 1192.34003
[8] Romanovski, V. G.; Robnik, M., The centre and isochronicity problems for some cubic systems, Journal of Physics A: Mathematical and General, 34, 47, 10267-10292 (2001) · Zbl 1014.34028 · doi:10.1088/0305-4470/34/47/326
[9] Romanovski, V. G.; Chen, X.; Hu, Z., Linearizability of linear systems perturbed by fifth degree homogeneous polynomials, Journal of Physics A: Mathematical and Theoretical, 40, 22, 5905-5919 (2007) · Zbl 1127.34020 · doi:10.1088/1751-8113/40/22/010
[10] Giné, J.; Romanovski, V. G., Integrability conditions for Lotka-Volterra planar complex quintic systems, Nonlinear Analysis: Real World Applications, 11, 3, 2100-2105 (2010) · Zbl 1194.34003 · doi:10.1016/j.nonrwa.2009.06.002
[11] Chen, X.; Romanovski, V. G., Linearizability conditions of time-reversible cubic systems, Journal of Mathematical Analysis and Applications, 362, 2, 438-449 (2010) · Zbl 1192.34037 · doi:10.1016/j.jmaa.2009.09.013
[12] Cairó, L.; Chavarriga, J.; Giné, J.; Llibre, J., Class of reversible cubic systems with an isochronous center, Computers and Mathematics with Applications, 38, 11, 39-53 (1999) · Zbl 0982.34024
[13] Gasull, A.; Guillamon, A.; Mañosa, V., An explicit expression of the first Liapunov and period constants with applications, Journal of Mathematical Analysis and Applications, 211, 1, 190-212 (1997) · Zbl 0882.34040 · doi:10.1006/jmaa.1997.5455
[14] Loud, W. S., Behavior of the period of soutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3, 21-36 (1964) · Zbl 0139.04301
[15] Pleshkan, I., A new method of investigating the isochronicity of system of two differential equations, Differential Equations, 5, 796-802 (1969) · Zbl 0252.34034
[16] Du, C.; Liu, Y.; Mi, H., A class of ninth degree system with four isochronous centers, Computers and Mathematics with Applications, 56, 10, 2609-2620 (2008) · Zbl 1165.34338 · doi:10.1016/j.camwa.2008.05.024
[17] Du, C. X.; Mi, H. L.; Liu, Y. R., Center, limit cycles and isochronous center of a Z4-equivariant quintic system, Acta Mathematica Sinica, 26, 6, 1183-1196 (2010) · Zbl 1214.34025 · doi:10.1007/s10114-010-7541-9
[18] Li, F.; Wang, M., Bifurcation of limit cycles in a quintic system with ten parameters, Nonlinear Dynamics, 71, 213-222 (2013) · Zbl 1268.34060
[19] Liu, Y.; Li, J., Periodic constants and time-angle difference of isochronous centers for complex analytic systems, International Journal of Bifurcation and Chaos, 16, 12, 3747-3757 (2006) · Zbl 1140.34301 · doi:10.1142/S0218127406017142
[20] Liu, Y., Theory of center-focus for a class of higher-degree critical points and infinite points, Science in China A: Mathematics, Physics, Astronomy, 44, 3, 365-377 (2001) · Zbl 1012.34027 · doi:10.1007/BF02878718
[21] Amel’kin, V. V.; Lukashevich, N. A.; Sadovskii, A. P., Nonlinear Oscillations in Second Order Systems (1982), Minsk, Belarus: Belarusian State University, Minsk, Belarus · Zbl 0526.70024
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.