Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples

Abstract

We implement a computer-assisted approach that, under appropriate conditions, allows the bifurcation analysis of the coarse dynamic behavior of microscopic simulators without requiring the explicit derivation of closed macroscopic equations for this behavior. The approach is inspired by the so-called time-step per based numerical bifurcation theory. We illustrate the approach through the computation of both stable and unstable coarsely invariant states for Kinetic Monte Carlo models of three simple surface reaction schemes. We quantify the linearized stability of these coarsely invariant states, perform pseudo-arclength continuation, detect coarse limit point and coarse Hopf bifurcations and construct two-parameter bifurcation diagrams.