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Over the last few decades, complex oscillations of slow-fast systems havebeen a key area of research. In the theory of slow-fast systems, the locationof singular Hopf bifurcation and maximal canard is determined by computing thefirst Lyapunov coefficient. In particular, the analysis of canards is based onthe genericity condition that the first Lyapunov coefficient must be non-zero.This manuscript aims to further extend the results to the case where the firstLyapunov coefficient vanishes. For that, the analytic expression of the secondLyapunov coefficient and the investigation of the normal form for codimension-2singular Bautin bifurcation in a predator-prey system is done by explicitlyidentifying the locally invertible parameter-dependent transformations. Aplanar slow-fast predator-prey model with Holling type III functional responseis considered here, where the prey population growth is affected by the weakAllee effect, and the prey reproduces much faster than the predator. Usinggeometric singular perturbation theory, normal form theory of slow-fastsystems, and blow-up technique, we provide a detailed mathematicalinvestigation of the system to show a variety of rich and complex nonlineardynamics including but not limited to the existence of canards, relaxationoscillations, canard phenomena, singular Hopf bifurcation, and singular Bautinbifurcation. Additionally, numerical simulations are conducted to support thetheoretical findings.

Normal form for singular Bautin bifurcation in a slow-fast system with Holling type III functional response