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We address a numerical methodology for the computation of coarse-grainedstable and unstable manifolds of saddle equilibria/stationary states ofmultiscale/stochastic systems for which a "good" macroscopic description in theform of Ordinary (ODEs) and/or Partial differential equations (PDEs) does notexplicitly/ analytically exists in a closed form. Thus, the assumption is thatwe have a detailed microscopic simulator of a complex system in the form ofMonte-Carlo, Brownian dynamics, Agent-based models e.t.c. (or a black-boxlarge-scale discrete time simulator) but due to the inherent complexity of theproblem, we don't have explicitly an accurate model in the form of ODEs orPDEs. Our numerical scheme is a three-tier one including: (a) the "on demand"detection of the coarse-grained saddle equilibrium, (b) its coarse-grainedstability analysis, and (c) the parametrization of the semi-local invariantstable and unstable manifolds by the numerical solution of thehomological/functional equations for the coefficients of the truncated seriesapproximation of the manifolds.

A Numerical Method for the Parametrization of Stable and Unstable Manifolds of Microscopic Simulators