# Exact Response Theory for Delay Equations

## TL;DR

This paper extends a method for predicting system responses to perturbations to include systems with delayed dynamics.

## Contribution

The paper adapts exact response theory for time-lagged systems using an augmented phase space approach.

## Key findings

- The delay-dependent dynamics can be mapped into an autonomous system in an augmented phase space.
- The comparison between linear and exact response approaches is explored for a specific kernel choice.

## Abstract

The exact response theory, also known as Transient Time Correlation Function formalism, is a powerful method concerning how observables respond to a given perturbation of the dynamics of the systems of interest, and it extends linear response theory to generic (autonomous) dynamical systems. Its main ingredient is the so-called dissipation function. In this paper, we adapt this theory for time-lagged systems, and we illustrate its applicability considering simple examples of delay equations, with different memory terms. Adopting the technique already used for time deterministic as well as stochastic time-dependent perturbations, the dynamics is described in a higher dimensional phase space, in which the delay-dependent dynamics is mapped into an augmented phase space: the new dynamics is proven to be autonomous and suitable for the exact responses to be computed. In addition, we explore the comparison between linear and exact approaches for a specific kernel choice.