Analytical theory for intermediate-frequency forcing

Develop a general analytical theory for the maximal Lyapunov exponent of periodically forced ecological networks in the intermediate-frequency regime, thereby providing a unified description of the crossover between the adiabatic and high-frequency regimes and a quantitative account of dynamical stabilization.

Background

The paper derives an analytical approximation for the maximal Lyapunov exponent under the adiabatic assumption of slowly varying environmental forcing and separately describes the asymptotic behavior produced by rapidly oscillating forcing. Numerical simulations show systematic deviations from the adiabatic prediction as the forcing frequency increases, including a high-frequency stabilization or rescue effect.

The unresolved gap concerns the intermediate-frequency regime, where neither the quasi-static approximation nor the high-frequency limiting argument is sufficient. The paper identifies Floquet theory, averaging methods, and Magnus expansions as possible tools for deriving a theory that quantitatively captures the crossover and the observed stabilization.

References

Finally, the analytical approximation developed here relies on the adiabatic assumption of slowly varying environmental forcing and therefore accurately describes the low-frequency regime. Although our analysis and numerical results also elucidate the asymptotic behavior in the high-frequency limit, a general analytical theory describing the intermediate-frequency regime remains unavailable.

— Periodic Environmental Forcing Shapes the Stability of Complex Ecological Networks  (2608.14081 - Chowdhury, 14 Aug 2026) in Section V, Discussion