Maximum admissible time interval for numerically computing cumulative interaction strength (CIS)

Determine the largest time interval τ for which the numerical computation of the cumulative interaction strength (CIS)—defined via the differential quotient of the continuous-time flow map with respect to infinitesimal perturbations of the initial state, and equivalent to the interaction Jacobian minus the identity—remains accurate and numerically stable in ordinary differential equation systems, particularly under chaotic dynamics.

Background

Within the ordinary differential equation framework, the paper defines a cumulative interaction strength (CIS) as a parametric interaction strength exactly equivalent to the interaction Jacobian, computed from the flow over a finite time interval by comparing perturbed and unperturbed trajectories.

For systems exhibiting chaotic dynamics, small initial perturbations can grow exponentially, which may compromise the numerical stability and accuracy of CIS when the evaluation horizon τ is large. The authors note that this raises uncertainty about how large τ can be chosen while still ensuring reliable CIS estimates and refer to supporting discussion in Appendix 4 on numerical stability.