Dynamics and geometric desingularization of the multiple time scale Fitzhugh Nagumo Rinzel model with fold singularity

The properties of dynamics in the fold region of the FitzHugh Nagumo Rinzel model is investigated in this paper. Firstly, the fast subsystem separated from this model by the slow-fast analysis method is verified to be bistable and the condition of bistability is also given. This implies that the model is bistable. Secondly, the critical manifold of this model is found to be non-horizontal and the hyperbolicity is lost at the so-called fold points |v|=1 where the singularities arise. In this case, the blowup method is applied to eliminate the singularity and the dynamics will be analyzed. With regard to the non-horizontal critical manifold, a new technique for choosing the blowup chart is proposed. Consequently, the complexity of the blowup method when dealing with the model with non-horizontal critical is reduced. Finally, the double mixed model oscillations, double canards and the feature transitions of the model are obtained numerically.