Evidence of dispersion relations for the nonlinear response of the Lorenz 63 system

TL;DR

This paper provides the first numerical evidence that dispersion relations and sum rules from nonlinear response theory apply to the Lorenz 63 system, confirming the theory's validity for chaotic non-equilibrium systems.

Contribution

It demonstrates the applicability of Kramers-Kronig dispersion relations to the Lorenz 63 system and introduces recursive methods for calculating harmonic susceptibilities in chaotic systems.

Findings

Numerical simulations agree with theoretical dispersion relations.

Dispersion relations hold for linear and second harmonic responses.

The results support extending nonlinear response theory to other non-equilibrium systems.

Abstract

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.