Detection of coarse-grained unstable states of microscopic/stochastic systems: a timestepper-based iterative protocol

Abstract

We address an iterative procedure that can be used to detect coarse-grained hyperbolic unstable equilibria (saddle points) of microscopic simulators when no equations at the macroscopic level are available. The scheme is based on the concept of coarse timestepping (Kevrekidis et al. in Commun. Math. Sci. 1(4):715–762, 2003) incorporating an adaptive mechanism based on the chord method allowing the location of coarse-grained saddle points directly. Ultimately, it can be used in a consecutive manner to trace the coarse-grained open-loop saddle-node bifurcation diagrams of complex dynamical systems and large-scale systems of ordinary and/or partial differential equations. We illustrate the procedure through two indicative examples including (i) a kinetic Monte Carlo simulation (kMC) of simple surface catalytic reactions and (ii) a simple agent-based model, a financial caricature which is used to simulate the dynamics of buying and selling of a large population of interacting individuals in the presence of mimesis. Both models exhibit coarse-grained regular turning points which give rise to branches of saddle points.