Singular Bautin bifurcation analysis of a slow–fast predator–prey system

Over the past few decades, the study of complex oscillations in slow–fast systems has been a focal point of research. Within the realm of slow–fast systems theory, the determination of the singular Hopf bifurcation and maximal canard locations relies on computing the first Lyapunov coefficient, based on the genericity condition that it is nonzero. This manuscript seeks to broaden these results to scenarios where the first Lyapunov coefficient becomes zero. To achieve this, the analytical expression of the second Lyapunov coefficient is derived, and an exploration of the normal form for codimension-2 singular Bautin bifurcation in a singularly perturbed Holling type-III predator–prey system with a weak Allee effect is conducted, explicitly identifying locally invertible parameter-dependent transformations. Utilizing geometric singular perturbation theory, the normal form theory of slow–fast systems, and the blow-up technique, this research unveils a diverse array of rich and complex nonlinear dynamics. This includes, but is not limited to, the existence of relaxation oscillations, canard phenomena, singular Hopf bifurcation, singular Bautin bifurcation, and saddle-node bifurcation of the canard cycle. Furthermore, numerical simulations are carried out to substantiate the theoretical findings.