Asymptotic behavior of the two-dimensional uncertainty exponent under domain expansion

Determine whether the uncertainty exponent u2 of the basin boundary subset Σ ∩ S2′ approaches 1 as the bounds on x1 and x2 are expanded outward from S2′ in the asymmetrically electrically coupled system of two identical non-chaotic Rulkov neurons with parameters σ1=σ2=−0.5, α1=α2=4.5, and coupling strengths g1e=0.05 and g2e=0.25, where S2′ = {X: −2 < x1 < 2, y1 = −3.25, −2 < x2 < 2, y2 = −3.25} and Σ is the boundary separating the basins of the non-chaotic spiking attractor and the chaotic spiking-bursting pseudo-attractor.

Background

The paper analyzes basin boundaries between two coexisting long-lived behaviors in an asymmetrically electrically coupled pair of identical non-chaotic Rulkov neurons: a non-chaotic spiking attractor and a chaotic spiking-bursting pseudo-attractor. The authors use the uncertainty exponent method to quantify final-state sensitivity of the basin boundary subsets in two- and four-dimensional regions.

For the two-dimensional slice S2′ defined by fixing y1=y2=−3.25 and varying x1,x2 in [−2,2], they compute an uncertainty exponent u2≈0.314 indicating fractal basin boundaries and significant final-state sensitivity. Based on basin classification showing dominance of the chaotic basin far from the attractors in this slice, they conjecture that u2 tends toward 1 when the domain expands beyond S2′, reflecting diminishing uncertainty in the limit of large x1,x2 ranges.

This open problem asks for a rigorous determination of the limiting behavior of u2 under domain expansion, clarifying whether final-state uncertainty vanishes asymptotically (u2→1) in the two-dimensional slice as the bounds on x1 and x2 grow.