Riddled Basins of Attraction in Systems Exhibiting Extreme Events

TL;DR

This paper investigates riddled basins of attraction in delay-coupled FitzHugh-Nagumo systems, revealing how initial conditions influence the emergence of extreme events and characterizing basin boundaries.

Contribution

It demonstrates the presence of riddled basins in such systems and introduces a method to quantify their complexity using the uncertainty exponent.

Findings

Riddled basins occur as coupling increases.

Pure regions avoid extreme events; mixed regions may produce them.

Tiny perturbations in mixed regions can trigger extreme events.

Abstract

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 the riddled basin 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.