Hopf and zero-Hopf bifurcations for a class of cubic Kolmogorov systems
in R3
- Qinlong Wang,
- Jingping Lu,
- Chunyong Wang,
- Wentao Huang
Wentao Huang
Central China Normal University School of Mathematics and Statistics
Author ProfileAbstract
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 2023Submission Checks Completed
27 Sep 2023Assigned to Editor
06 Oct 2023Review(s) Completed, Editorial Evaluation Pending
08 Oct 2023Reviewer(s) Assigned