Bounded random perturbations in chaotic systems with intermingled basins of attraction

Abstract

This work investigates how bounded modeling uncertainties affect systems with intermingled basins. We analyze two systems, a modified Kan map and a coupled Lorenz-like system, both of which exhibit intermingled basin structures. A bounded random perturbation, representing modeling uncertainty, is introduced, and proximity to invariant sets is measured through a numerical tolerance. By varying the perturbation over twelve orders of magnitude and considering different tolerance values, we show that the dynamics are primarily organized by the ratio between perturbation amplitude and classification tolerance, rather than by their absolute values. For small ratios, trajectories remain near invariant sets; for intermediate values, residence times collapse; and for large ratios, trajectories are expelled from invariant neighborhoods. In the coupled Lorenz-like system, synchronized states decay smoothly as uncertainty increases, whereas antisynchronized states break down abruptly. Overall, both models display the same sequence of dynamical regimes, suggesting that this sensitivity mechanism is robust across different dynamical structures.