Stability of traveling waves involving turning points

In this paper, we consider a FitzHugh-Nagumo type equation, the critical manifolds of the corresponding traveling equation of which admit turning points. The simple pulse is a perturbation of a singular orbit consisting of two fast jumps and two slow motions. We show that the stability of traveling pulse relies on how the jump back arises, i.e., how the unstable manifold of the right critical manifold meets the stable manifold of the left one. Specially, we show that the pulse is stable if the sign of the derivative $ a^\prime(w_b) $ at the $ w- $value $ w_b $ of the back is positive, i.e., $ a^\prime(w_b)>0 $ and unstable if $ a^\prime(w_b)<0 $.