# A theory of average response to large jump perturbations

## TL;DR

This paper extends the Fluctuation Dissipation theorem to large, instantaneous jumps in nonlinear dynamical systems, showing that average responses can still be expressed via time correlation functions of unperturbed dynamics, including stochastic cases.

## Contribution

It develops a theoretical framework for understanding the average response of nonlinear systems to large, instantaneous perturbations, both deterministic and stochastic, using correlation functions.

## Key findings

- Average response formulas derived for large jumps
- Applicable to stochastic and deterministic jumps
- Efficient approximations for practical use

## Abstract

A key feature of the classical Fluctuation Dissipation theorem is its ability to approximate the average response of a dynamical system to a sufficiently small external perturbation from an appropriate time correlation function of the unperturbed dynamics of this system. In the present work, we examine the situation where the state of a nonlinear dynamical system is perturbed by a finitely large, instantaneous external perturbation (jump) -- for example, the Earth climate perturbed by an extinction level event. Such jump can be either deterministic or stochastic, and in the case of a stochastic jump its randomness can be spatial, or temporal, or both. We show that, even for large instantaneous jumps, the average response of the system can be expressed in the form of a suitable time correlation function of the corresponding unperturbed dynamics. For stochastic jumps, we consider two situations: one where a single spatially random jump of a system state occurs at a predetermined time, and another where jumps occur randomly in time with small space-time dependent statistical intensity. For all studied configurations, we compute the corresponding average response formulas in the form of suitable time correlation functions of the unperturbed dynamics. Some efficiently computable approximations are derived for practical modeling scenarios.