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Hopf and zero-Hopf bifurcations for a class of cubic Kolmogorov systems in R3
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  • Qinlong Wang,
  • Jingping Lu,
  • Chunyong Wang,
  • Wentao Huang
Qinlong Wang
Guilin University of Electronic Technology

Corresponding Author:[email protected]

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Jingping Lu
Guilin University of Electronic Technology
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Chunyong Wang
Hezhou University
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Wentao Huang
Central China Normal University School of Mathematics and Statistics
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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 that there exist at most seven small amplitude limit cycles restricted to one center manifold and Hopf cyclicity 8 at the positive equilibrium. Secondly, by using the normal form algorithm, we discuss the existence of stable periodic solution 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 is rarely considered. Finally, some related illustrations are given by means of numerical simulation.
27 Sep 2023Submitted to Mathematical Methods in the Applied Sciences
27 Sep 2023Assigned to Editor
27 Sep 2023Submission Checks Completed
06 Oct 2023Review(s) Completed, Editorial Evaluation Pending
08 Oct 2023Reviewer(s) Assigned