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

TL;DR

This paper introduces a numerical method to compute stable and unstable manifolds of saddle points in complex multiscale systems using microscopic simulators without explicit macroscopic models.

Contribution

It presents a three-tier numerical scheme for detecting equilibria, analyzing stability, and parametrizing invariant manifolds directly from microscopic simulators.

Findings

Effective detection of saddle equilibria in complex systems.

Numerical parametrization of invariant manifolds without explicit macroscopic equations.

Applicable to Monte Carlo, agent-based, and other black-box simulators.

Abstract

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.