  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.
