Linear response of the Lorenz system

Abstract

The present numerical study provides strong evidence that at standard parameters the response of the Lorenz system to small perturbations of the control parameter r is linear. This evidence is obtained not only directly by determining the response in the observable A(x)=z, but also indirectly by validating various implications of the assumption of a linear response, like a quadratic response at twice the perturbation frequency, a vanishing response in A(x)=x, the Kramers-Kronig relations, and relations between different response functions. Since the Lorenz system is nonhyperbolic, the present results indicate that in contrast to a recent speculation the large system limit (thermodynamic limit) need not be invoked to obtain a linear response for chaotic systems of this type.