Riddled basins of attraction in systems exhibiting extreme events.

Using a system of two FitzHugh-Nagumo units, we demonstrate the occurrence of riddled basins of attraction in delay-coupled systems as the coupling between the units is increased. We characterize riddled basins using the uncertainty exponent which is a measure of the dimensions of the basin boundary. Additionally, we show that the phase space can be partitioned into pure and mixed regions, where initial conditions in the pure regions certainly avoid the generation of extreme events, while initial conditions in the mixed region may or may not exhibit such events. This implies that any tiny perturbation of initial conditions in the mixed region could yield the emergence of extreme events because the latter state possesses a riddled basin of attraction.