In this paper, we give a detailed study of the stable region in discrete-time FitzHugh–Nagumo delayed excitable Systems, which can be divided into two parts: one is independent of delay and the other is dependent on delay. Two different new states are to be observed, which are new steady states (equilibria-the excitable FitzHugh–Nagumo) or limit cycles/higher periodic orbits (the FitzHugh–Nagumo oscillators) as the origin loses its stability, and usually, one is synchronized and the other asynchronized. We also find out that there exist critical curves through which there occur fold bifurcations, flip bifurcations, Neimark–Sacker bifurcations and even higher-codimensional bifurcations etc. It is also shown that delay can play an important role in rich dynamics, such as the occurrence of chaos or not, by means of Lyapunov exponents, Lyapunov dimensions, and the sensitivity to the initial conditions. Multistability phenomena are also discussed including the coexistence of synchronized and asynchronized oscillators, or synchronized/asynchronized oscillators and multiple stable nontrivial equilibria etc.
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