Linear response, susceptibility and resonances in chaotic toy models
Abstract
We consider simple examples illustrating some new features of the linear response theory developed by Ruelle for dissipative and chaotic systems [D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Stat. Phys. 95 (1999) 393468]. In this theory the concepts of linear response, susceptibility and resonance, which are familiar to physicists, have been revisited due to the dynamical contraction of the whole phase space onto attractors. In particular the standard framework of the “fluctuationdissipation” theorem breaks down and new resonances can show up outside the power spectrum. In previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by Ruelle in a model of chaotic neural networks. In this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the “stable” resonances and their consequences on the linear response of the system. We consider a class of twodimensional timediscrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics θ=2θ_{t}(mod2π). A generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies [B. Cessac, J.A. Sepulchre, Stable resonances and signal propagation in a chaotic network of coupled units, Phys. Rev. E 70 (2004) 056111; B. Cessac, J.A. Sepulchre, Transmitting a signal by amplitude modulation in a chaotic network, Chaos 16 (2006) 01310411310412]. These models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions. Also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies, widths, and dependence on the pair of perturbed/observed variables.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 January 2007
 DOI:
 10.1016/j.physd.2006.09.034
 arXiv:
 arXiv:nlin/0612026
 Bibcode:
 2007PhyD..225...13C
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Dynamical Systems
 EPrint:
 36 pages, 11 Figures, to appear in Physica D