arXiv subjects

Kuilin Wu

Publications and source records attributed to Kuilin Wu.

3 recordsLinked to original sources

arXiv2025

Bifurcation analysis for a SIRS model with a nonlinear incidence rate

In this paper, the main purpose is to explore an SIRS epidemic model with a general nonlinear incidence rate $f(I)S=βI(1+\upsilon I^{k-1})S$ ($k>0$). We analyzed the existence and stability of equilibria of the epidemic model. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms of the epidemic model are derived for different types of bifurcation, including Bogdanov-Takens bifurcation, Nilpotent focus bifurcation and Hopf bifurcation. The first four focal values are computed to determine the codimension of the Hopf bifurcation, which can be undergo some limit cycles. Some numerical results and simulations are presented to illustrate these theoretical results.

math.DS↗

arXiv2025

A Degenerate Bifurcation Perspective on High Sensitivity in a Modified Gower-Leslie Model with Additive Allee Effect

The population dynamics in a modified Leslie-Gower model with an additive Allee effect are highly sensitive to both parameters and initial population densities, leading to outcomes ranging from coextinction to sustained multistable steady states. This work links this sensitivity to complicated bifurcations. We establish the existence of a codimension 4 nilpotent cusp and a corresponding degenerate Bogdanov-Takens bifurcation with codimension 4, which critically shape the system's response to parameter changes. Most significantly, we prove that the Hopf bifurcation occurring at a center-type equilibrium can give rise to up to five limit cycles-a phenomenon scarcely documented in previous ecological studies-thereby inducing a pronounced dependence of oscillatory regimes on initial conditions. Numerical simulations confirming heteroclinic loops and multiple limit cycles provide consistent support for the theoretical analysis.

math.DS↗

arXiv2024

On the stability of singular Hopf bifurcation and its application

Recently, research on the complex periodic behavior of multi-scale systems has become increasingly popular. Krupa et al. \cite{krupa2} provided a way to obtain relaxation oscillations in slow-fast systems through singular Hopf bifurcations and canard explosion. The authors derived a $O(1)$ expression $A$ for the first Lyapunov coefficient (under the condition $A \neq 0$), and deduced the bifurcation curves of singular Hopf and canard explosions. This paper employs Blow-up technique, normal form theory, and Lyapunov coefficient formula to present higher-order approximate expressions for the first Lyapunov coefficient when $A=0$ for slow-fast systems. As an application, we investigate the bifurcation phenomena of a predator-prey model with Allee effects. Utilizing the formulas obtained in this paper, we identify both supercritical and subcritical Hopf bifurcations that may occur simultaneously in the system. Numerical simulations validate the results. Finally, by normal form and slow divergence integral theory, we prove the cyclicity of the system is 1.

math.DS↗