Stability of the four-dimensional uncertainty exponent under domain expansion

Ascertain whether the uncertainty exponent u4 of the basin boundary subset Σ ∩ S4′ remains approximately constant as the bounds of the four-dimensional region are expanded beyond S4′ 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 S4′ = {X: −2 < x1 < 2, −1 < y1 < −5, −2 < x2 < 2, −1 < y2 < −5} and Σ is the boundary separating the basins of the non-chaotic spiking attractor and the chaotic spiking-bursting pseudo-attractor.

Background

In four-dimensional state space, the authors classify both basins as Class 2, meaning they occupy fixed fractions of state space. For the hypercube S4′ near the attractors, they compute an extremely small uncertainty exponent u4≈0.037, indicating near-maximal fractality of the basin boundary and extreme final-state sensitivity.

Given the Class 2 nature of both basins in the full four-dimensional space, the authors conjecture that u4 remains relatively constant when the domain expands beyond S4′, reflecting persistent overlap between the basins throughout the enlarged region.

This open problem seeks to determine the robustness of u4 under domain enlargement, thereby clarifying whether extreme final-state sensitivity persists across scales in four-dimensional state space for this coupled Rulkov neuron system.