Linear Response and Optimal Fingerprinting for Nonautonomous Systems

We provide a link between response theory, pullback measures, and optimal fingerprinting method that paves the way for a) predicting the impact of acting forcings on time-dependent systems and b) attributing observed anomalies to acting forcings when the reference state in not time-independent. We first derive formulas for linear response theory for time-dependent Markov chains and diffusions processes. We discuss existence, uniqueness, and differentiability of the pullback measure under general (not necessarily slow or periodic) perturbations of the transition kernels. An explicit Green-Kubo-type formula for the linear response is derived. We analyze in detail the case of periodic reference dynamics, where the unperturbed pullback attractor is periodic but the response is generally not. Our formulas reduce to those of classic linear response if one considers a reference autonomous state. Finally, we show that our results allow for extending the theory of optimal fingerprinting for detection and attribution of climate change (or change in any complex system) for the case of time-dependent background state and for the case where the optimal solution is sought for multiple time slices at the same time. We provide strong numerical support for the findings by applying our theory to a modified version of the Ghil-Sellers energy balance model where we include explicit time dependence in the reference state as a result of natural forcings. We verify the accuracy of response theory in predicting the impact of increases of $CO_2$ in the temperature field even when we discretize the system using Markov state modelling approach. Additionally, we consider a more complex modelling scenario where a localized aerosol forcing is also included in the system and show that the optimal fingerprinting method developed here is able to attribute the climate change signal to the acting forcings.

💡 Research Summary

This paper establishes a comprehensive linear response framework for non‑autonomous (time‑dependent) dynamical systems and demonstrates how the same theory can be used to extend optimal fingerprinting (OFM) for detection and attribution (D&A) problems when the background state is itself evolving. The authors first treat two mathematically distinct but related settings: finite‑state time‑dependent Markov chains and continuous‑time diffusion processes. Under a uniform Dobrushin contraction assumption they prove the existence, uniqueness and differentiability of the pullback (or non‑autonomous) invariant measure, which replaces the usual stationary invariant measure in autonomous theory.

For a perturbed transition kernel (M^{\varepsilon}_n = M_n + \varepsilon m_n) they derive an explicit first‑order expression for the perturbed pullback measure: \