HOPF AND ZERO-HOPF BIFURCATIONS FOR A CLASS OF CUBIC KOLMOGOROV SYSTEMS IN $ \mathbb R^{3}$

Jingping Lu; Chunyong Wang; Wentao Huang; Qinlong Wang

doi:10.11948/20240108

2025 Volume 15 Issue 1

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Jingping Lu, Chunyong Wang, Wentao Huang, Qinlong Wang. HOPF AND ZERO-HOPF BIFURCATIONS FOR A CLASS OF CUBIC KOLMOGOROV SYSTEMS IN $ \mathbb R^{3}$[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 354-372. doi: 10.11948/20240108

Citation:

Jingping Lu, Chunyong Wang, Wentao Huang, Qinlong Wang. HOPF AND ZERO-HOPF BIFURCATIONS FOR A CLASS OF CUBIC KOLMOGOROV SYSTEMS IN $ \mathbb R^{3}$[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 354-372. doi: 10.11948/20240108

HOPF AND ZERO-HOPF BIFURCATIONS FOR A CLASS OF CUBIC KOLMOGOROV SYSTEMS IN $ \mathbb R^{3}$

1.
School of Mathematics and Computational Science, School of Computer Science and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China
2.
School of Mathematics and Statistics, Guilin University of Technology, Guilin 541004, China
3.
College of Science, Hezhou University, Hezhou 542899, China

Author Bio: Email: lujingbaby520@163.com(J. Lu); Email: wangchunyong@163.com(C. Wang); Email: huangwentao@163.com(W. Huang)
Corresponding author: Email: wqinlong@163.com(Q. Wang)
Fund Project: The authors were supported by National Natural Science Foundation of China (12161023, 12061016) and the Basic Ability Enhancement Program for Young and Middle-aged Teachers of Guangxi (No. 2022KY0254) and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Center for Applied Mathematics of Guangxi (GUET) and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation

Abstract

Abstract

In this paper, Hopf and zero-Hopf bifurcations are investigated for a class of three-dimensional cubic Kolmogorov systems with one positive equilibrium. Firstly, by computing the singular point quantities and figuring out center conditions, we determined that the highest order of the positive equilibrium is eight as a fine focus, which yields Hopf cyclicity eight at the positive equilibrium. Secondly, by extending the normal form method, we discuss the existence of multiple periodic solutions via zero-Hopf bifurcation around the positive equilibrium. At the same time, the relevance between zero-Hopf bifurcation and Hopf bifurcation is analyzed via its special case, which are rarely studied in detail.
- Three-dimensional Kolmogorov system /
- Hopf bifurcations /
- zero-Hopf bifurcations /
- center manifold /
- center problem
MSC: 34C23, 34C28, 34C40, 37G15

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References

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