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A common approach to studying high-dimensional systems with emergentlow-dimensional behavior is based on lift-evolve-restrict maps (calledequation-free methods): first, a user-defined lifting operator maps a set oflow-dimensional coordinates into the high-dimensional phase space, then thehigh-dimensional (microscopic) evolution is applied for some time, and finallya user-defined restriction operator maps down into a low-dimensional spaceagain. We prove convergence of equation-free methods for finite time-scaleseparation with respect to a method parameter, the so-called healing time. Ourconvergence result justifies equation-free methods as a tool for performinghigh-level tasks such as bifurcation analysis on high-dimensional systems.  More precisely, if the high-dimensional system has an attracting invariantmanifold with smaller expansion and attraction rates in the tangentialdirection than in the transversal direction (normal hyperbolicity), andrestriction and lifting satisfy some generic transversality conditions, then animplicit formulation of the lift-evolve-restrict procedure generates anapproximate map that converges to the flow on the invariant manifold forhealing time going to infinity. In contrast to all previous results, our resultdoes not require the time scale separation to be large. A demonstration withMichaelis-Menten kinetics shows that the error estimates of our theorem aresharp.  The ability to achieve convergence even for finite time scale separation isespecially important for applications involving stochastic systems, where theevolution occurs at the level of distributions, governed by the Fokker-Planckequation. In these applications the spectral gap is typically finite. Weinvestigate a low-dimensional stochastic differential equation where the ratiobetween the decay rates of fast and slow variables is 2.

Convergence of equation-free methods in the case of finite time scale separation with application to deterministic and stochastic systems