Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity

We prove the existence of multiple periodic solutions as well as the presence of complex profiles (for a certain range of the parameters) for the steady state solutions of a class of reaction diffusion equations with a FitzHugh-Nagumo cubic type nonlinearity. An application is given to a second order ODE related to a myelinated nerve axon model.