  We address a numerical methodology for the computation of coarse-grained
stable and unstable manifolds of saddle equilibria/stationary states of
multiscale/stochastic systems for which a "good" macroscopic description in the
form of Ordinary (ODEs) and/or Partial differential equations (PDEs) does not
explicitly/ analytically exists in a closed form. Thus, the assumption is that
we have a detailed microscopic simulator of a complex system in the form of
Monte-Carlo, Brownian dynamics, Agent-based models e.t.c. (or a black-box
large-scale discrete time simulator) but due to the inherent complexity of the
problem, we don't have explicitly an accurate model in the form of ODEs or
PDEs. Our numerical scheme is a three-tier one including: (a) the "on demand"
detection of the coarse-grained saddle equilibrium, (b) its coarse-grained
stability analysis, and (c) the parametrization of the semi-local invariant
stable and unstable manifolds by the numerical solution of the
homological/functional equations for the coefficients of the truncated series
approximation of the manifolds.
