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

TL;DR

This paper introduces an iterative, timestepper-based method to detect unstable coarse-grained saddle points in microscopic stochastic systems without explicit macroscopic equations, demonstrated on chemical and financial models.

Contribution

It presents a novel adaptive iterative protocol for locating coarse-grained saddle points directly from microscopic simulators, enabling bifurcation analysis without macroscopic equations.

Findings

Successfully identified saddle points in kinetic Monte Carlo simulations.

Applied the method to agent-based financial models with observable turning points.

Demonstrated tracing of bifurcation diagrams in stochastic systems.

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., 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. We illustrate the procedure through two indicatively 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.