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Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained\/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological\/functional equations for the coefficients of a polynomial approximation of the manifolds.<\/jats:p>","DOI":"10.1007\/s11075-021-01155-0","type":"journal-article","created":{"date-parts":[[2021,7,9]],"date-time":"2021-07-09T05:02:26Z","timestamp":1625806946000},"page":"1335-1368","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A numerical method for the approximation of stable and unstable manifolds of microscopic simulators"],"prefix":"10.1007","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9568-3355","authenticated-orcid":false,"given":"Constantinos","family":"Siettos","sequence":"first","affiliation":[]},{"given":"Lucia","family":"Russo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,7,9]]},"reference":[{"key":"1155_CR1","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/j.ymssp.2015.12.039","volume":"84","author":"DA Barton","year":"2017","unstructured":"Barton, D.A.: Control-based continuation: bifurcation and stability analysis for physical experiments. 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