HOPF BIFURCATION PROBLEM FOR A CLASS OF KOLMOGOROV MODEL WITH A POSITIVE NILPOTENT CRITICAL POINT

Chaoxiong Du; Wentao Huang

doi:10.11948/20210276

2022 Volume 12 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2022 > 12(4): 1451-1465

Next Article Previous Article

Chaoxiong Du, Wentao Huang. HOPF BIFURCATION PROBLEM FOR A CLASS OF KOLMOGOROV MODEL WITH A POSITIVE NILPOTENT CRITICAL POINT[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1451-1465. doi: 10.11948/20210276

Citation:

Chaoxiong Du, Wentao Huang. HOPF BIFURCATION PROBLEM FOR A CLASS OF KOLMOGOROV MODEL WITH A POSITIVE NILPOTENT CRITICAL POINT[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1451-1465. doi: 10.11948/20210276

HOPF BIFURCATION PROBLEM FOR A CLASS OF KOLMOGOROV MODEL WITH A POSITIVE NILPOTENT CRITICAL POINT

Chaoxiong Du^1, ,,
Wentao Huang²

1.
School of Mathematics, Changsha Normal University, Changsha, Hunan, 410100, China
2.
College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, Guangxi, China

Corresponding author: Email: ducx123@126.com(C. Du)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 12061016) and the Research Fund of Hunan provincial education department(No. 18A525)

Abstract

Abstract

In this paper, We discuss the Hopf bifurcation problem of a three-order positive nilpotent critical point (1, 1) of a class of Kolmogorov model. By using the method offered by [12], we obtain the expressions of quasi-Lyapunov constants with the help of computer algebra system-MATHEMATICA. By analyzing the structure of these quasi-lyapunov constants, we divide them into two kinds of cases and study their bifurcation behavior separately. For case 1, the nilpotent critical point (1, 1) can bifurcate 5 small amplitude limit cycles. For case 2, 6 small amplitude limit cycles can bifurcate from the three-order nilpotent critical point (1, 1). In addition, We also give the integrability conditions (i.e., center condition) for each case. In terms of limit cycle bifurcation for Kolmogorov model with nilpotent positive critical points, our result is new.
- Hopf bifurcation /
- nilpotent critical point /
- kolmogorov model /
- quasiLyapunov constant
MSC: 34C07, 34C23

FullText(HTML)

References

[1]	D. Addona, L. Angiuli and L. Lorenzi, Invariant measures for systems of Kolmogorov equations, J. Appl. Anal. Comput., 2018, 8(3), 764–804. Google Scholar
[2]	C. Du and W. Huang, Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model, Nonlinear Dyn., 2013, 72(1), 197–206. Google Scholar
[3]	C. Du, Y. Liu and W. Huang, Limit cycles bifurcations for a class of Kolmogorov model in symmetrical vector field, Int. J. Bifurcation and Chaos., 2014, 24(4), 1450040. Google Scholar
[4]	C. Du, Y. Liu and W. Huang, Limit cycles bifurcations behavior for a class of quartic Kolmogorov model in symmetrical vector field, Appl. Math. Model., 2016, 40(5–6), 4094–4108. doi: 10.1016/j.apm.2015.11.029 CrossRef Google Scholar
[5]	C. Du, Q. Wang and W. Huang, Three-Dimensional hopf bifurcation for a class of cubic Kolmogorov system, Int. J. Bifurcation and Chaos., 2014, 24(3), 1450036. doi: 10.1142/S0218127414500369 CrossRef Google Scholar
[6]	M. Han and V. G. Romanovski, On the number of limit cycles of polynomial Lienard systems, Nonlinear Anal. Real., 2013, 14, 1655–1668. doi: 10.1016/j.nonrwa.2012.11.002 CrossRef Google Scholar
[7]	D. He, W. Huang and Q. Wang, Small Amplitude Limit Cycles and Local Bifurcation of Critical Periods for a Quartic Kolmogorov System, Qual. Theor. Dyn. Syst., 2020, 68(19). Google Scholar
[8]	X. Huang and L. Zhu, Limit cycles in a general kolmogorov model, Nonlinear Anal. Real., 2005, 60, 1394–1414. Google Scholar
[9]	J. Llibre, Y. Paulina Martsnez and C. Valls, Limit cycles bifurcating of Kolmogorov systems in R² and in R³, Commun. Nonlinear Sci., 2020, 91, 105401. doi: 10.1016/j.cnsns.2020.105401 CrossRef Google Scholar
[10]	N. G. Lloyd etc. A cubic Kolmogorov system with six limit cycles, Comput. Math. Appl., 2002, 44(3–4), 445–455. doi: 10.1016/S0898-1221(02)00161-X CrossRef Google Scholar
[11]	Y. Liu and J. Li, On the study of three-order nilpotent critical points: integral factor method, Int. J. Bifurcation and Chaos., 2011, 21(5), 497–504. Google Scholar
[12]	Y. Liu and J. Li, Bifurcations of limit cycles created by a multiple nilpotent critical point of planar dynamical systems, Int. J. Bifurcation and Chaos., 2011, 21(2), 1293–1309. Google Scholar
[13]	E. Saez and I. Szanto, Limit cycles of a cubic kolmogorov system, Appl. Math. Lett., 1996, 9(1), 15–18. doi: 10.1016/0893-9659(95)00095-X CrossRef Google Scholar
[14]	J. Wang and W. Jiang, Bifurcation and chaos of a delayed predator-prey model with dormancy of predators, Nonlinear Dynam., 2012, 69, 1541–1558. doi: 10.1007/s11071-012-0368-4 CrossRef Google Scholar
[15]	Y. Wu, M. Zhang and J. Mao, Bifurcations of limit cycles at a nilpotent critical point in a septic Lyapunov system, J. Appl. Anal. Comput., 2020, 10(6), 2575–2591. Google Scholar
[16]	Y. Yuan etc., The limit cycles of a general Kolmogorov system, J. Math. Anal. Appl., 2012, 392(2), 225–237. doi: 10.1016/j.jmaa.2012.02.065 CrossRef Google Scholar