Chaotic oscillations in singularly perturbed FitzHugh–Nagumo systems

We consider the singularly perturbed limit of periodically excited two-dimensional FitzHugh–Nagumo systems. We show that the dynamics of such systems are essentially governed by a one-dimensional map and present a numerical scheme to accurately compute it together with its Lyapunov exponent. We then investigate the occurrence of chaos by varying the parameters of the system, with especial emphasis on the simplest possible chaotic oscillations. Our results corroborate and complement some recent works on bifurcations and routes to chaos in certain particular cases corresponding to piecewise linear FitzHugh–Nagumo-like systems.