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.

我们提出了一种近似多尺度/随机系统的鞍平衡/静止状态的粗粒度稳定和不稳定流形的数值方法，对于这些系统的宏观描述不以封闭形式分析存在。因此，潜在的假设是我们有一个详细的微观模拟器（蒙特卡罗、分子动力学、基于代理的模型等），它描述了复杂系统（或黑盒大型模拟器）的子单元的动力学，但是我们没有明确可用的封闭形式的动力学模型来描述新兴的粗粒度/宏观动力学。我们的数值方案基于无方程多尺度框架，它是一个三层程序，包括（a）粗粒度鞍平衡的收敛，(b) 其粗粒度稳定性分析，以及 (c) 局部不变稳定和不稳定流形的近似；后面的任务是通过对流形多项式近似系数的一组同调/函数方程的数值解来实现的。