Center-manifold extension of the adiabatic-elimination method

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Abstract
A generic situation for the bifurcation of stationary states is considered. The separation of the relaxation time scales is such that the adiabatic-elimination method is incompatible with the linearization of the reduced equation of motion around the unstable steady state. Thus, the system is assumed to have departed considerably from the critical point. The functional dependence of the fast-relaxing degree of freedom on the order parameter is obtained by finding the nonadiabatic center-manifold equation. 2 This equation is an extension of the adiabatic-elimination procedure since it holds even in the case of a small separation of time scales. The results are applied to a system studied extensively where the reduced equation of motion admits a bistable viscous potential. The macroscopic equations of motion are compatible with the obtained stochastic description.