Autocorrelations in the center manifold of dissipative systems

The phase-space contraction for the systems under consideration corresponds to an asymptotic-center manifold functional dependence of the fast-relaxing variables, Xf,js, on the slowly-relaxing ones, the Xs,is. The propagation of perturbations along the center manifold is obtained from the Green-function sensitivity matrix. Scaling and self-similarity relations among the matrix elements are found. This fact allows for a simplification in the computation of the Xf,j-Xf,j and the Xs,i-Xs,i autocorrelations. The validity of the results is confirmed in two different contexts: (a) An analytical derivation of power spectra at the onset of periodic instabilities is performed. We demonstrate that in the infinite relaxation time limit for slow variables, the power spectra for all the Xf,js converge to the same distribution. This result is in accord with previous computations. (b) The power spectrum for a randomly driven anharmonic damped oscillator is computed at the asymptotic-center manifold regime and tested vis-a$aa-vis previous plots exhibiting a very good agreement.