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.