一类具有十参数的复三次多项式微分系统的弱持续中心问题

doi:10.16088/j.issn.1001-6600.2025090101

Abstract

Abstract: The problem of weakly persistent centers for a ten-parameter family of complex planar cubic polynomial differential systems at the origin is studied. Firstly, the necessary conditions for the origin of the system to be a weakly persistent center are obtained by calculating and decomposing the algebraic variety of the ideal generated by the first seven focal quantities. Then, it is proved that these conditions are also sufficient by either using Darboux integrable theory to construct the Darboux first integral or Darboux integrating factor, or verifying the time reversibility of the system, or applying the induction to demonstrate the existence of polynomial first integral. Finally, the complete classifications for a weakly persistent center in a family of real planar cubic polynomial differential systems is derived, which is obtained by setting the variables and the coefficients of the complex systems are conjugation.

Key words: cubic polynomial differential systems, center, weakly persistent center, focal quantity, Darboux integrable theory

CLC Number: O175

Luo Cheng, Huang Wentao, He Dongping, Zhang Yue. Weakly persistent centers for a ten-parameter family of complex planar cubic polynomial differential systems[J].Journal of Guangxi Normal University(Natural Science Edition), 2026, 44(4): 121-129.

References

[1] Li J B. Hilbert’s 16th problem and bifurcations of planar polynomial vector fields[J]. International Journal of Bifurcation and Chaos, 2003, 13(1): 47-106. DOI: 10.1142/s0218127403006352.
[2] 姚洁, 王勤龙. 一类三维系统高阶奇点退化Hopf分岔的环性[J]. 广西师范大学学报(自然科学版), 2026, 44(1): 102-109.DOI: 10.16088/j.issn.1001-6600.2025012401.
[3] 刘桔坤, 黄文韬, 刘宏普. 一类三维三次系统极限环的新下界[J]. 广西师范大学学报(自然科学版), 2022, 40(6): 109-115. DOI: 10.16088/j.issn.1001-6600.2021102002.
[4] 张二丽, 邢玉清. 具有不变直线的非Hamilton系统的极限环分支[J]. 广西师范大学学报(自然科学版), 2020, 38(3): 45-51. DOI: 10.16088/j.issn.1001-6600.2020.03.006.
[5] Bautin N N. On The number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type[J]. American Mathematical Society Translations, 1954, 100(1): 1-19.
[6] Malkin K E. Criteria for the center for a certain differential equation[J]. Volzhskii Matem Sbornik, 1964, 2: 87-91.
[7] Lloyd N G, Pearson J M, Saéz E, et al. A cubic Kolmogorov system with six limit cycles[J]. Computers & Mathematics with Applications, 2002, 44(3/4): 445-455. DOI: 10.1016/S0898-1221(02)00161-X.
[8] Pearson J M, Lloyd N G. Kukles revisited: advances in computing techniques[J]. Computers & Mathematics with Applications, 2010, 60(10): 2797-2805. DOI: 10.1016/j.camwa.2010.09.034.
[9] Sadovskii A P. Solution of the center and focus problem for a cubic system of nonlinear oscillations[J]. Differentsial’nye Uravneniya, 1997, 33(2): 236-244.
[10] Liu Y R, Li J B. Complete study on a bi-center problem for the Z₂-equivariant cubic vector fields[J]. Acta Mathematica Sinica, English Series, 2011, 27(7): 1379-1394. DOI: 10.1007/s10114-011-8412-8.
[11] Romanovski V G, Fernandes W, Oliveira R. Bi-center problem for some classes of Z₂-equivariant cubic vector fields[J]. Journal of Computational and Applied Mathematics, 2017, 320: 61-75. DOI: 10.1016/j.cam.2017.02.003.
[12] Dukarić M, Fernandes W, Oliveira R. Symmetric centers on planar cubic differential systems[J]. Nonlinear Analysis, 2020, 197: 111868. DOI: 10.1016/j.na.2020.111868.
[13] Li F, Wu Y S, Yu P. Complete classification on center of cubic planar systems symmetric with respect to a straight line[J]. Communications in Nonlinear Science and Numerical Simulation, 2023, 120: 107167. DOI: 10.1016/j.cnsns.2023.107167.
[14] Cima A, Gasull A, Medrado J C. On persistent centers[J]. Bulletin Des Sciences Mathématiques, 2009, 133(6): 644-657. DOI: 10.1016/j.bulsci.2008.08.007.
[15] Chen X W, Romanovski V G, Zhang W N. Persistent centers of complex systems[J]. Bulletin Des Sciences Mathématiques, 2014, 138(1): 110-123. DOI: 10.1016/j.bulsci.2013.10.003.
[16] Christopher C, Mardeić P, Rousseau C. Normalizable, integrable, and linearizable saddle points for complex quadratic systems in C²[J]. Journal of Dynamical and Control Systems, 2003, 9(3): 311-363. DOI: 10.1023/A:1024643521094.
[17] Christopher C, Rousseau C. Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system[J]. Qualitative Theory of Dynamical Systems, 2004, 5(1): 11-61. DOI: 10.1007/BF02968129.
[18] Romanovski V G, Subǎ A. Centers of some cubic systems[J]. Annals of Differential Equations, 2001, 17(4): 363-376. DOI: CNKI:SUN:WFFN.0.2001-04-006.
[19] Dukarić M, Giné J. Integrability of Lotka-Volterra planar complex cubic systems[J]. International Journal of Bifurcation and Chaos, 2016, 26(1): 1650002. DOI: 10.1142/s0218127416500024.
[20] Romanovskii V G, Shcheglova N L. The integrability conditions for two cubic vector fields[J]. Differential Equations, 2000, 36(1): 108-112. DOI: 10.1007/BF02754169.
[21] Dukarić M. On integrability and cyclicity of cubic systems[J]. Electronic Journal of Qualitative Theory of Differential Equations, 2020(55): 1-19. DOI: 10.14232/ejqtde.2020.1.55.
[22] Liu Y R, Li J B, Huang W T. Planar dynamical systems: selected classical problems[M]. Beijing: Science Press, 2014. DOI: 10.1515/9783110298369.
[23] Shafer D, Romanovski V. The center and cyclicity problems: a computational algebra approach[M]. Boston: Birkhäuser, 2009. DOI: 10.1007/978-0-8176-4727-8.
[24] Chavarriga J, Giacomini H, Giné J, et al. On the integrability of two-dimensional flows[J]. Journal of Differential Equations, 1999, 157(1): 163-182. DOI: 10.1006/jdeq.1998.3621.
[25] Ferec B, Giné J. A blow-up method to prove formal integrability for some planar differential systems[J]. Journal of Applied Analysis and Computation, 2018, 8(6): 1833-1850. DOI: 10.11948/2018.1833.

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Weakly persistent centers for a ten-parameter family of complex planar cubic polynomial differential systems

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