Bifurcation of limit cycles and isochronous centers on center manifolds for four-dimensional systems.(English)Zbl 1531.34045
This paper studies the bifurcation of limit cycles and isochronous centers on center manifolds for four-dimensional nonlinear dynamic systems. These systems under study have a pair of purely imaginary eigenvalues and a pair of eigenvalues with negative real part for the Jacobian matrix corresponding to the singularity situated at the origin. By computing singular point values at the origin, some interesting results are obtained and the article is well written.
Reviewer: Yanqin Xiong (Nanjing)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
Keywords:
center manifold;four-dimensional system;isochronous center;isochronous constant;limit cycleCite
Full Text:DOI
References:
| [1] | Hilbert, D., Mathematical problems, Bull Am Math Soc, 8, 10, 437-479, 1902 ·JFM 33.0976.07 |
| [2] | Ilyashenko, Y., Centennial history of Hilbert’s 16th problem, Bull Am Math Soc, 39, 03, 301-355, 2002 ·Zbl 1004.34017 |
| [3] | Li, J., Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int J Bifur Chaos, 13, 01, 47-106, 2003 ·Zbl 1063.34026 |
| [4] | Wang, Q.; Liu, Y.; Chen, H., Hopf bifurcation for a class of three‐dimensional nonlinear dynamic systems, Bull Sci Math, 134, 7, 786-798, 2010 ·Zbl 1204.37051 |
| [5] | Tian, Y.; Yu, P., Seven limit cycles around a focus point in a simple three‐dimensional quadratic vector field, Int J Bifur Chaos, 24, 06, 2014 ·Zbl 1296.34101 |
| [6] | Yu, P.; Han, M., Ten limit cycles around a center‐type singular point in a 3‐d quadratic system with quadratic perturbation, Appl Math Lett, 44, 17-20, 2015 ·Zbl 1336.34051 |
| [7] | Wang, Q.; Huang, W.; Feng, J., Multiple limit cycles and centers on center manifolds for Lorenz system, Appl Math Comput, 238, 281-288, 2014 ·Zbl 1334.37089 |
| [8] | Huang, W.; Wang, Q.; Chen, A., Hopf bifurcation and the centers on center manifold for a class of three‐dimensional circuit system, Math Methods Appl Sci, 43, 4, 1988-2000, 2020 ·Zbl 1453.34069 |
| [9] | Sang, B.; Wang, Q.; Huang, W., Computation of focal values and stability analysis of 4‐dimensional systems, Electron J Differ Equ, 209, 1-11, 2015 ·Zbl 1329.34070 |
| [10] | Llibre, J.; Makhlouf, A.; Badi, S., 3‐dimensional Hopf bifurcation via averaging theory of second order, Discr Contin Dyn Syst, 25, 4, 1287-1295, 2009 ·Zbl 1186.37059 |
| [11] | Silva, P.; Buzzi, C.; Llibre, J., 3‐dimensional Hopf bifurcation via averaging theory, Discr Contin Dyn Syst, 17, 529-540, 2017 ·Zbl 1137.37026 |
| [12] | Edneral, VF; Mahdi, A.; Romanovskic, VG; Shafer, DS, The center problem on a center manifold in \(\operatorname{\mathbb{R}}^3\), Nonlinear Anal Real World Appl, 75, 2614-2622, 2012 ·Zbl 1259.34021 |
| [13] | Romanovski, V.; Shafer, D., Computation of focus quantities of three dimensional polynomial systems, J Shanghai Norm Univ (Natural Sciences), 43, 5, 529-544, 2014 |
| [14] | Buica, A.; García, IA; Maza, S., Multiple Hopf bifurcation in \(\mathbb{R}^3\) and inverse Jacobi multipliers, J Differ Equ, 256, 310-325, 2014 ·Zbl 1346.37050 |
| [15] | García, IA; Maza, S.; Shafer, DS, Cyclicity of polynomial nondegenerate centers on center manifolds, J Differ Equ, 265, 11, 5767-5808, 2018 ·Zbl 1434.37030 |
| [16] | García, IA; Maza, S.; Shafer, DS, Center cyclicity of Lorenz, Chen and Lü systems, Nonlinear Anal, 188, 362-376, 2019 ·Zbl 1428.37046 |
| [17] | Needham, DJ, A centre theorem for two‐dimensional complex holomorphic systems and its generalizations, Proc R Soc Lond A, 450, 1939, 225-232, 1995 ·Zbl 0835.34007 |
| [18] | Loud, WS, Behavior of the period of solutions of certain plane autonomous systems near centers, Contrib Differ Equ, 3, 21-36, 1964 ·Zbl 0139.04301 |
| [19] | Pleshkan, I., A new method of investigating the isochronicity of a system of two differential equations, Differ Equ, 5, 796-802, 1969 ·Zbl 0252.34034 |
| [20] | Chavarriga, J.; Giné, J.; García, IA, Isochronous centers of a linear center perturbed by fourth degree homogrneous polynomial, Bull Sci Math, 123, 2, 77-96, 1999 ·Zbl 0921.34032 |
| [21] | Chavarriga, J.; Giné, J.; García, IA, Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomial, J Comput Appl Math, 126, 1‐2, 351-368, 2000 ·Zbl 0978.34028 |
| [22] | Dolićanin‐Dekić, D., Strongly isochronous centers of cubic systems with degenerate infinity, Differ Equ, 50, 7, 971-975, 2014 ·Zbl 1316.34030 |
| [23] | Llibre, J.; Romanovski, VG, Isochronicity and linearizability of planar polynomial Hamiltonian systems, J Differ Equ, 259, 5, 1649-1662, 2015 ·Zbl 1347.37103 |
| [24] | Li, F.; Liu, Y.; Liu, Y.; Yu, P., Complex isochronous centers and linearization transformations for cubic Z \({}_2\)‐equivariant planar systems, J Diff Eqs, 268, 7, 3819-3847, 2020 ·Zbl 1511.34041 |
| [25] | Chicone, C.; Jacobs, M., Bifurcations of critical periods for planar vector fields, Trans Amer Math Soc, 312, 2, 433-486, 1989 ·Zbl 0678.58027 |
| [26] | Gasull, A.; Guillamon, A.; Mañosa, V., An explicit expression of the first Liapunov and period constants with applications, J Math Anal Appl, 211, 1, 190-212, 1997 ·Zbl 0882.34040 |
| [27] | Gasull, A.; Guillamon, A.; Mañosa, V., The period function for Hamiltonian systems with homogeneous nonlinearities, J Math Anal Appl, 139, 237-260, 1997 ·Zbl 0891.34033 |
| [28] | Yu, P.; Han, M., Critical periods of planar revertible vector field with third‐degree polynomial functions, Int J Bifur Chaos, 19, 19-433, 2009 ·Zbl 1170.34316 |
| [29] | Romanovski, VG; Shafer, AS, The center and cyclicity problems, A Computational Algebra Approach, 2009, Boston: Birkhäuser ·Zbl 1192.34003 |
| [30] | Christopher, CJ; Devlin, J., Isochronous centers in planar polynomial systems, SIAM J Math Anal, 28, 162-177, 1997 ·Zbl 0881.34057 |
| [31] | Liu, Y.; Huang, W., A new method to determine isochronous center conditions for polynomial differential systems, Bull Sci Math, 127, 2, 133-148, 2003 ·Zbl 1034.34032 |
| [32] | Huang, W.; Chen, A.; Xu, Q., Bifurcation of limit cycles and isochronous centers for a quartic system, Int J Bifur Chaos, 10, 2013 ·Zbl 1277.34044 |
| [33] | Chen, T.; Huang, W.; Ren, D., Weak centers and local critical periods for a \(Z_2\)‐equivariant cubic system, Nonlinear Dyn, 78, 2319-2329, 2014 |
| [34] | Wang, Q.; Huang, W.; Du, C., The isochronous center on center manifolds for three dimensional differential systems, 2019 |
| [35] | Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential system, Sci China (Ser a), 33, 10-24, 1990 ·Zbl 0686.34027 |
| [36] | Amel’kin, VV; Lukashevich, NA; Sadovsky, AP, Non‐linear Oscillations in the Systems of Second Order, 1982, Minsk: Belarusian University Press ·Zbl 0526.70024 |
| [37] | Carr, J., Applications of Centre Manifold Theory, 1981, New York: Springer‐Verlag ·Zbl 0464.58001 |
| [38] | Yu, P.; Corless, R., Symbolic computation of limit cycles associated with Hilbert’s 16th problem, Commun Nonlin Sci Numer Simulat, 14, 12, 4041-4056, 2009 ·Zbl 1221.34082 |
| [39] | Han, M.; Yang, J., The maximum number of zeros of functions with parameters and application to differential equations, J Nonlinear Model Anal, 3, 13-34, 2021 |
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.
