Small amplitude limit cycles and local bifurcation of critical periods for a quartic Kolmogorov system.(English)Zbl 1446.34050
Summary: In this paper, small amplitude limit cycles and the local bifurcation of critical periods for a quartic Kolmogrov system at the single positive equilibrium point \((1,1)\) are investigated. Firstly, through the computation of the singular point values, the conditions of the critical point \((1,1)\) to be a center and to be the highest degree fine singular point are derived respectively. Then, we prove that the maximum number of small amplitude limit cycles bifurcating from the equilibrium point \((1,1)\) is 7. Furthermore, through the computation of the period constants, the conditions of the critical point \((1,1)\) to be a weak center of finite order are obtained. Finally, we determine the number of local critical periods bifurcating from the equilibrium point \((1,1)\) under the center conditions. It is the first example of a quartic Kolmogorov system with seven limit cycles and a quartic Kolmogorov system with four local critical periods created from a single positive equilibrium point.
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of 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 |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
Cite
Full Text:DOI
References:
| [1] | Amel’kin, VV; Lukashevich, NA; Sadovskii, AP, Nonlinear Oscillations in Second Order Systems (1982), Minsk: Belarusian State University, Minsk ·Zbl 0526.70024 |
| [2] | Chen, X.; Zhang, W., Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232, 565-581 (2009) ·Zbl 1178.13015 ·doi:10.1016/j.cam.2009.06.029 |
| [3] | Chen, X.; Huang, W.; Romanovski, VG; Zhang, W., Linearizability and local bifurcation of critical periods in a cubic Kolmogorov system, J. Comput. Appl. Math., 245, 86-96 (2013) ·Zbl 1280.34033 ·doi:10.1016/j.cam.2012.12.003 |
| [4] | 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) ·doi:10.1007/s11071-014-1560-5 |
| [5] | Chicone, C.; Jacobs, M., Bifurcation of critical periods for plane vector fields, Trans. Am. Math. Soc., 312, 433-486 (1989) ·Zbl 0678.58027 ·doi:10.1090/S0002-9947-1989-0930075-2 |
| [6] | Du, C.; Huang, W., Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model, Nonlinear Dyn., 72, 197-206 (2013) ·Zbl 1269.92065 ·doi:10.1007/s11071-012-0703-9 |
| [7] | Du, C.; Liu, Y.; Huang, W., Limit cycles bifurcations for a class of Kolmogorov model in symmetrical vector field, Int. J. Bifur. Chaos, 24, 746-753 (2014) ·Zbl 1296.34095 |
| [8] | Du, C.; Liu, Y.; Huang, W., Behavior of limit cycle bifurcations for a class of quartic Kolmogorov models in a symmetrical vector field, Appl. Math. Model., 40, 4094-4108 (2016) ·Zbl 1459.34084 ·doi:10.1016/j.apm.2015.11.029 |
| [9] | Du, C.; Liu, Y.; Mi, H., The bifurcation of limit cycles for a class of cubic Kolmogorov system (in Chinese), Chin. J. Eng. Math., 24, 746-752 (2007) ·Zbl 1140.34351 |
| [10] | Du, C.; Liu, Y.; Zhang, Q., Limit cycles in a class of quartic Kolmogorov model with three positive equilibrium points, Int. J. Bifur. Chaos, 25, 17 (2015) ·Zbl 1317.34036 ·doi:10.1142/S0218127415500807 |
| [11] | Han, M.; Lin, Y.; Yu, P., A study on the exitence of limit cycles of a planar system with third-degree polynomials, Int. J. Bifur. Chaos, 14, 41-60 (2004) ·Zbl 1078.34017 ·doi:10.1142/S0218127404009247 |
| [12] | Huang, W.; Liu, Y., Bifurcations of limit cycles from infinity for a class of quintic polynomial system, Bull. Sci. Math., 128, 291-301 (2004) ·Zbl 1070.34064 ·doi:10.1016/j.bulsci.2004.02.002 |
| [13] | Huang, W.; Chen, T.; Li, J., Isolated periodic wave trains and local critical wave lengths for a nonlinear reaction-diffusion equation, Commun. Nonlinear. Sci. Numer. Simulat., 74, 84-96 (2019) ·Zbl 1464.35148 ·doi:10.1016/j.cnsns.2019.03.003 |
| [14] | Lloyd, NG; Pearson, JM; Sáez, E.; Szántó, I., A cubic Kolmogorov system with six limit cycles, Comput. Math. Appl., 44, 445-455 (2002) ·Zbl 1210.34048 ·doi:10.1016/S0898-1221(02)00161-X |
| [15] | Li, F., Integrability and bifurcations of limit cycles in a cubic Kolmogorov system, Int. J. Bifur. Chaos, 23, 1350061 (2013) ·Zbl 1270.34043 ·doi:10.1142/S0218127413500612 |
| [16] | Li, J., Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos, 13, 47-106 (2003) ·Zbl 1063.34026 ·doi:10.1142/S0218127403006352 |
| [17] | Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33, 10-23 (1990) ·Zbl 0686.34027 |
| [18] | Liu, Y.; Chen, H., Formulas of singurlar point quantities and the first \(10\) saddle quantities for a class of cubic system (in Chinese), Acta Math. Sin., 25, 295-302 (2002) ·Zbl 1014.34021 |
| [19] | Liu, Y.; Huang, W., A new method to determine isochronous center conditions for polynomial differential systems, Bull. Sci. Math., 127, 133-148 (2003) ·Zbl 1034.34032 ·doi:10.1016/S0007-4497(02)00006-4 |
| [20] | Liu, Y.; Li, J.; Huang, W., Singular Point Values, Center Problem and Bifurcations of Limit Cycles of Two Dimensional Differential Autonomous Systems (2008), Beijing: Science Press, Beijing |
| [21] | Liu, Y.; Li, J., New study on the center problem and bifurcations of limit cycles for the Lyapunov system (I ), Int. J. Bifur. Chaos, 19, 3791-3801 (2009) ·Zbl 1182.34044 ·doi:10.1142/S0218127409025110 |
| [22] | Lin, Y.; Li, J., The canonical form of the autonomous planar system and the critical point of the closed orbit period (in Chinese), Acta Math. Sin., 34, 490-501 (1991) ·Zbl 0744.34041 |
| [23] | Mi, H.; Du, C., The central conditidn and bifurcation of limit cycles for a class of cubic Kolmogorov system, Math. Theory Appl., 25, 19-21 (2005) ·Zbl 1504.34095 |
| [24] | Romanovski, VG; Han, M., Critical period bifurcations of a cubic system, J. Phys. A: Math. Gen., 36, 5011-5022 (2003) ·Zbl 1037.34034 ·doi:10.1088/0305-4470/36/18/306 |
| [25] | Romanovski, VG; Fernandes, W.; Tang, Y.; Tian, Y., Linearizability and critical period bifurcations of a generalized Riccati system, Nonlinear Dyn., 90, 257-269 (2017) ·Zbl 1390.70056 ·doi:10.1007/s11071-017-3659-y |
| [26] | Rousseau, C.; Toni, B., Local bifurcations of critical periods in vector fields with homogeneous nonliearities of the third degree, Can. J. Math., 36, 473-484 (1993) ·Zbl 0792.58030 ·doi:10.4153/CMB-1993-063-7 |
| [27] | Rousseau, C.; Toni, B., Local bifurcations of critical periods in the reduced Kukles system, Can. J. Math., 49, 338-358 (1997) ·Zbl 0885.34033 ·doi:10.4153/CJM-1997-017-4 |
| [28] | Toni, B., Bifurcations of critical periods: cubic vector fields in Kapteyns normal form, Quaestiones Mathematicae, 22, 43-61 (1999) ·Zbl 0938.34023 ·doi:10.1080/16073606.1999.9632058 |
| [29] | Xu, Q.; Huang, W., The center conditions and local bifurcation of critical periods for a Liénard system, Appl. Math. Comput., 217, 6637-6643 (2011) ·Zbl 1218.34034 |
| [30] | Ye, Y.; Ye, W., Cubic Kolmogorov differential system with two limit cycles surrounding the same focus, Ann. Diff. Eqs., 1, 201-207 (1985) ·Zbl 0597.34020 |
| [31] | Yu, P.; Han, M., Critical periods of planar revertible vector field with third-degree polynomial functions, Int. J. Bifur. Chaos, 19, 419-433 (2009) ·Zbl 1170.34316 ·doi:10.1142/S0218127409022981 |
| [32] | Yu, P.; Han, M.; Zhang, J., Critical periods of third-order planar Hamiltonian systems, Int. J. Bifur. Chaos, 20, 2213-2224 (2010) ·Zbl 1196.34049 ·doi:10.1142/S0218127410027040 |
| [33] | Wu, Y.; Huang, W.; Suo, Y., Weak center and bifurcation of critical periods in a cubic \(Z_2\)-equivariant Hamiltonian vector field, Int. J. Bifur. Chaos, 25, 1550143 (2015) ·Zbl 1327.34055 ·doi:10.1142/S0218127415501436 |
| [34] | Wu, D.; Huang, W.; Wu, Y., Limit cycles of a cubic kolmogorov system (in Chinese), J. Guilin Univ. Electron. Technol., 36, 160-163 (2016) |
| [35] | Wu, D.; Huang, W.; Wu, Y., Bifurcation of limit cycles of a class of quartic Kolmogorov system (in Chinese), J. Henan Univ. Sci. Technol. (Nat. Sci.), 37, 382-86 (2016) ·Zbl 1363.34108 |
| [36] | Zhan, J.; Huang, W.; He, D., The center and limit cycles of a quartic Kolmogorov system (in Chinese), J. Guilin Univ. Electron. Technol., 38, 242-246 (2018) |
| [37] | Zhang, Q.; Li, F.; Zhao, Y., Limit cycles in a cubic kolmogorov system with harvest and two positive equilibrium points, Abstr. Appl. Anal., 2014, 1-6 (2014) ·Zbl 1474.92136 |
| [38] | Zhang, W.; Hou, X.; Zeng, Z., Weak centres and bifurcation of critical periods in reversible cubic systems, Comput. Math. Appl., 40, 771-782 (2000) ·Zbl 0962.34025 ·doi:10.1016/S0898-1221(00)00195-4 |
| [39] | Zou, L.; Chen, X.; Zhang, W., Local bifurcations of critical periods for cubic Liénard equations with cubic damping, J. Comput. Appl. Math., 222, 404-410 (2008) ·Zbl 1163.34349 ·doi:10.1016/j.cam.2007.11.005 |
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.
