Many physical problems, most importantly the quantification of climate change, involve estimating the response of a deterministic chaotic dynamical system's statistical equilibrium to perturbations in the dynamics. If the equilibrium varies differentiably with the perturbation, it is possible to compute a Taylor expansion of this response from only the unperturbed dynamics of the system. Many practitioners working with high-dimensional complex systems assume the differentiability requirement holds, at least for "large-scale" observables: however, theoretical work has shown that it fails for even very simple chaotic systems such as the logistic map. To understand this discrepancy, we analyse model systems consisting of large weakly-coupled networks of chaotic subsystems that may individually have a non-differentiable response. We derive reduced large-scale dynamics, and show that under physically reasonable assumptions the response of the large-scale observables is differentiable.
(Joint work with Georg Gottwald)