Evidence of Dispersion Relations for the Nonlinear Response of the Lorenz 63 System |
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Authors: | Valerio Lucarini |
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Institution: | (1) Department of Physics, INFN, University of Bologna, Viale Berti Pichat 6/2, Bologna, Italy |
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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 derive asymptotic behaviors and 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. |
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Keywords: | Lorenz system Non-equilibrium steady states Ruelle response theory Kramers-Kronig relations Sum rules Axiom A Singular hyperbolic system Harmonic generation Susceptibility Numerical simulation Time series analysis Spectrum Periodic forcing Climate |
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