# Linear repsonse theory

A central recent development in the field of chaotic dynamical systems is the theory of linear response. This theory determines whether the expectation value of an observable of a dynamical system changes differentiably upon changing a parameter of the dynamics. Furthermore, the theory gives a formula for the value of the derivative with respect to the parameter (the celebrated fluctuation-dissipation theorem is an example of such a formula). Such theories have been well known in statistical mechanics community in the form of the fluctuation-dissipation theorem for stochastic systems. The proof of linear response for deterministic hyperbolic dynamical systems is however a relatively recent achievement due to Ruelle (Ruelle, 1997; Ruelle, 2004). However, it has recently been proved that for a large class of dynamical systems the invariant measure does not depend smoothly on the parameter and typical observables do not obey a linear response (Baladi, 2014).

The value of the linear response theory in a physical setting, and in particular the formulae determining the magnitude of the linear response, is evident. The FDT depends only on correlation functions of the unperturbed system, meaning that the linear response can be experimentally predicted without ever perturbing the system.

With Valerio Lucarini, Tobias Kuna and Davide Faranda, I have analysed
numerical methods for sampling the **linear response** of dynamical
systems (Lucarini et al., 2012). We have studied the efficiency of
different methods of numerically sampling the linear response functional. In
this work we have for the first time derived a sampling formula for
single-frequency periodic perturbations that had been previously proposed in
the literature on heuristic grounds. Perhaps more importantly, we have
demonstrated that this method cannot be used to sample linear response to a
perturbation with a continuous frequency spectrum. We have therefore proposed
a new sampling method that can sample such a response .

I have worked with Georg Gottwald (University of Sydney) and Caroline Wormell on dynamical systems with a non-differentiable response to perturbations. We have developed a sensitive statistical test to numerically determine whether or not the response to a perturbation is differentiable or not. This method is a vast improvement over the ad-hoc methods that have been used up until now in the climate sciences. Using this method and a theoretical analysis of the invariant measure we demonstrate that the detectability of non-differentiability crucially depends on the localisation and scale of the observable, the perturbation size and the sample size. We showed that scientists using a finite amount of data (which in the case of even simple systems such as the logistic map can be of the order of $10^6$) may be falsely led to conclude a linear response exists even if the dynamical system does not in fact obey linear response. We furthermore demonstrate that naively applying FDT to the non-differentiable system gives erroneous results.

- Ruelle, D. (1997). Differentiation of SRB States.
*Communications in Mathematical Physics*,*187*(1), 227–241. https://doi.org/10.1007/s002200050134 - Ruelle, D. (2004). Differentiation of SRB states for hyperbolic flows.
*Math/0408097*. - Baladi, V. (2014). Linear response, or else.
*ArXiv:1408.2937 [Math]*. - Lucarini, V., Kuna, T., Wouters, J., & Faranda, D. (2012). Relevance of sampling schemes in light of Ruelle’s
linear response theory.
*Nonlinearity*,*25*(5), 1311–1327. https://doi.org/10.1088/0951-7715/25/5/1311