Geometric investigation of low-dimensional manifolds in systems approaching equilibrium

Many systems approach equilibrium slowly along surfaces of dimension smaller than the original dimensionality. Such systems include coupled chemical kinetics and master equations. In the past the steady state approximation has been used to estimate these lower dimensional surfaces, commonly referred to as “manifolds,” and thus reduce the dimensionality of the system which needs to be studied. However, the steady state approximation is often inaccurate and sometimes difficult to define unambiguously. In recent years two methods have been proposed to go beyond the steady state approximation to improve the accuracy of dimension reduction. We investigate these methods and suggest significant modifications to one of them to allow it to be used for the generation of low-dimensional manifolds in large systems. Based on the geometric investigations, two other approaches are suggested which have some advantages over these two methods for the cases studied here. All four approaches are geometric and offer advantages over methods based on the evaluation of time-dependent behavior, where phenomenological rate laws are extracted from the time-dependent behavior.