Numerical approximation of center, stable and unstable manifolds of multiscale/stochastic systems

The numerical approximation of center, unstable and stable ivariant manifolds is important for a series of system-level tasks, particularly for the analysis and control large-scale systems. For example, embedding the high-dimensional dynamics on the center manifold allows the stability analysis and the control on a lower dimensional space. Here, we extend our previous work to address a numerical methodology for the computation of such manifolds for 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 dynamical simulator of a complex system in the form of Monte-Carlo, Brownian dynamics, Agent-based e.t.c. but we don’t have a explicitly a system of ODEs or PDEs in a closed form for the “slow” evolving variables. Based on this assumption, Gear & Kevrekidis and Gear et al. addressed an approach to compute “slow” manifolds by restricting the higher-order derivatives of the “fast” variables to zero. Our numerical scheme is a three-tier one including the (a) on ”demand” detection of the (coarse-grained) non-hyperbolic equilibrium, (b) stability analysis of the critical point(s), and (c) approximation of local invariant manifolds by identifying the nuemrical quantities required (residuals, Jacobians, Hessians, etc)