Existence of linear-scaling perturbations ensuring CIS convergence over finite horizons in chaotic ODEs

Establish whether, for continuous-time dynamical systems with chaotic attractors and any fixed evaluation horizon τ2 > 0, there always exists a sufficiently small perturbation magnitude Δx > 0 such that the difference between perturbed and unperturbed trajectories at time τ2 scales as O(Δx), thereby guaranteeing convergence of the finite-difference computation of cumulative interaction strength; if not, characterize when this condition fails.

Background

The paper formulates CIS numerically by taking the limit of the ratio of the trajectory difference at time τ to the perturbation size Δx. In chaotic systems, trajectory separations may grow rapidly, potentially violating linear scaling with Δx over finite horizons. The appendix frames a stability proposition: for any large τ2, ensure there exists a Δx small enough so that the trajectory difference remains O(Δx), avoiding divergence and guaranteeing that the finite-difference approximation converges.

The authors explicitly note uncertainty about whether this condition is always satisfied in the chaotic setting, which directly impacts the reliability of CIS estimation over larger horizons.