Delay-induced Multistability in a Generic Model for Excitable Dynamics

Motivated by real-world excitable systems such as neuron models and lasers, we consider a paradigmatic model for excitability with a global bifurcation, namely a saddle-node bifurcation on a limit cycle. We study the effect of a time-delayed feedback force in the form of the difference between a system variable at a certain time and at a delayed time. In the absence of delay the only attractor in the system in the excitability regime, below the global bifurcation, is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling as well as saddle-node bifurcations of limit cycles are found in accordance with Shilnikov’s theorems.