Fundamental sensitivity propagators in dissipative systems with a statistical enslaving of fast-relaxing variables

We consider dissipative systems where the contraction in phase space corresponds asymptotically to the center-manifold-functional dependence of the fast-relaxing variables Xfs on the order parameters given by the slow variables Xss. We derive analytically the Green-function sensitivity matrix to calculate the response on the subordinated variables to instantaneous perturbations on Xf or on Xs. We thus show how these perturbations propagate along the center manifold. This information is obtained from sensitivity measurements on the experimentally relevant variables Xss using also the center-manifold expansion. The results are applied to a randomly driven damped anharmonic oscillator. The spectral density is found by Fourier transforming the sum of the different contributions to the equilibrium correlation: The XsXs, XsXf, and XfXf terms. The average local attractivity of the center manifold is calculated and used in this computation. The sensitivity functions involving only the subordinated degrees of freedom are used as fundamental propagators to calculate the XfXf term. The results are tested against an analog computer simulation and vis-à-vis plots obtained from statistical linearization methods. A very good agreement is observed.