Abstract
We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. 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.
Highlights
The computation of invariant manifolds of dynamical systems is very important for a series of system-level tasks, for the bifurcation analysis and control
Krauskopf et al [27] addressed a numerical method for the approximation of two-dimensional stable and unstable manifolds which incorporates the solution of a boundary value problem; the method performs a continuation of a family of trajectories possessing the same arc-length
We address a new multiscale numerical method for the approximation of the local stable and unstable manifolds based on the equation-free framework
Summary
The computation of invariant manifolds of dynamical systems is very important for a series of system-level tasks, for the bifurcation analysis and control. Building up on a previous work for the computation of coarse-grained center manifolds of microscopic simulators [43], we present a new numerical method based on the equation-free approach for the computation of coarse-grained stable and unstable manifolds of saddles of microscopic dynamical simulators (and in general large-scale discrete-time black-box simulators). The first example is a simple toy discrete-time map and the second one is a Gillespie-Monte Carlo realization of a simple catalytic reaction scheme describing the dynamics of CO oxidation on catalytic surfaces This is the first work that addresses the equation-free computation of both coarse-grained stable and unstable manifolds of saddles of microscopic simulators.
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