Entrainment by periodic perturbations in the center manifold at Ginzburg-Landau critical regimes

We consider the effect of periodic perturbations on open reactive systems far from the linear thermodynamics domain. The systems present a center-manifold contraction of the phase space achieved by sustaining a hard-mode instability. The critical regime is governed by a Ginzburg-Landau potential defined on the locally attractive, locally invariant center manifold. Within the framework of this reduction scheme, scaling properties of the Green's function sensitivity matrix are obtained. It is demonstrated that the entrainment is produced by a projection of the perturbing time-dependent field on the center manifold. Thus, the reduced equations for entrainment in the order parameter space are derived. It is demonstrated that two inherent properties of the system favor the entrainment: (a) a small sensitivity of the amplitude of the bifurcating limit cycle with respect to changes in the control parameter, (b) the departure of the system from the region of marginal stability. The results are applied in two different contexts: When there exists a separation of relaxation-time scales (in a truncation of Hopf's model for hydrodynamic turbulence) and when there is only one time scale involved (Brusselator). Agreement with previous derivations of the entrainment regions is found. Finally, a realistic experiment coupling two oscillatory reactors is suggested in order to test the theoretical findings. In this case, the frequency of the perturbation is a function of the bifurcation control parameter (the residence time) which measures the departure of the entrained system from marginal stability. The results are applicable in the case of convection in a rotating layer and convection driven by the Soret-Dufour effect since the oscillatory convection corresponds to a center manifold contraction of the phase space. This manifold contains the dominant velocity modes when the frequency of oscillation is small.