Global Dynamical Behavior of FitzHugh–Nagumo Systems with Invariant Algebraic Surfaces

In this paper, we characterize the global dynamical behaviors of FitzHugh–Nagumo system $$\dot{x}=z$$ , $$\dot{y}=b(x-dy)$$ , $$\dot{z}=x(x-1)(x-a)+y+cz$$ which has invariant algebraic surfaces. As byproducts, we obtain some new dynamical phenomena related to the invariant surfaces at the infinity, which does not appear previously in the study of other models. In addition, since the system restricted to the invariant algebraic surfaces is not analytic, we adopt some new techniques to overcome this difficulty.