# Response and Sensitivity Using Markov Chains

## TL;DR

This paper introduces a methodology using transfer operators and Markov chain approximations to analyze the response and sensitivity of dynamical systems to perturbations, with applications to climate models.

## Contribution

It develops a novel approach combining transfer operator formalism and perturbation theory for Markov matrices to assess system sensitivity and response.

## Key findings

- Effective calculation of linear and nonlinear response.
- Application to climate response models.
- Framework for analyzing sensitivity in dynamical systems.

## Abstract

Dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties. A prominent real-life example of this class of problems is the investigation of climate response to perturbations. In this respect, it is crucial to determine what the linear response of a system is to small perturbations as a quantification of sensitivity. Alongside previous work, here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations. By projecting the transfer operator onto a suitable finite dimensional vector space, one is able to obtain matrix representations which determine finite Markov processes. Further, using perturbation theory for Markov matrices, it is possible to determine the linear and nonlinear response of the system given a prescribed forcing. Here, we suggest a methodology which puts the scope on the evolution law of densities (the Liouville/Fokker-Planck equation), allowing to effectively calculate the sensitivity and response of two representative dynamical systems.