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-stepper 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.

We present a computer-assisted study of ''coarse'' stability/bifurcation calculations for kinetic Monte Carlo simulators using the so-called coarse timestepper approach presented in A. G. Makeev, D. Maroudas, and I. G. Kevrekidis, J. Chem. Phys. 116, 10083 ͑2002͒. Our illustrative example is a model of a heterogeneous catalytic surface reaction with repulsive adsorbate-adsorbate interactions and fast diffusion. Through numerical continuation and stability analysis, we construct one-and two-parameter coarse bifurcation diagrams. We also discuss several computational issues that arise in the process, the most important of which is the ''lifting'' of coarse, macroscopic initial conditions ͑moments of adsorbate distributions͒ to fine, microscopic initial conditions ͑distributions conditioned on these moments͒.

Coarse timesteppers provide a bridge between microscopic / stochastic system descriptions and macroscopic tasks such as coarse stability/bifurcation computations. Exploiting this computational enabling technology, we present a framework for designing observers and controllers based on microscopic simulations, that can be used for their coarse control. The proposed methodology provides a bridge between traditional numerical analysis and control theory on the one hand and microscopic simulation on the other. Introduction.Mathematical models, whether identified on-line during an experiment or derived from first principles, constitute the backbone of modern control practice. Models of chemical and transport processes (material, energy and momentum balances) traditionally take the form of deterministic ordinary or partial differential/algebraic evolution equations for macroscopic variables (e.g. concentrations). An arsenal of mathematical and computational tools targeted at such macroscopic models has been developed over the years for the performance of macroscopic, system-level tasks such as temporal simulation, stability and bifurcation analysis, optimization, design and control. For many processes of current interest, however, the best available description of the physics (through molecular dynamics, MD, kinetic Monte Carlo, KMC, kinetic theory based Lattice-Boltzmann, LB, or Markov chain simulators) operates at a vastly different scale from that at which the questions of interest are asked and the answers are required (e.g. operating diagrams and controller design for expected reaction rates). The implication is that macroscopic rules (description at a higher level) can somehow be deduced from microscopic ones (description at a much finer level). In most current problems, however, ranging from ecology to materials science and from chemistry to engineering, the closures required to translate microscopic/stochastic models to a high-level, macroscopic description are simply not available. A

A mathematical model is presented which reproduces the experimental results of anisotropic front propagation in the bistable H2+O2 reaction on a Rh(110) surface. A model represented by a system of two coupled nonlinear reaction–diffusion equations incorporates the chemical diffusion of adsorbed hydrogen and oxygen. In previous experiments with a photoelectron emission microscope (PEEM) it had been demonstrated that in the system H2+O2/Rh(110) the front anisotropy varied strongly with the experimental parameters. Depending upon temperature and hydrogen partial pressure the reaction fronts were elongated in the [11̄0]-direction or in the [001]-direction of Rh(110). Key features of the mathematical model are diffusion of hydrogen and oxygen and the strong inhibitory site-blocking effect of adsorbed oxygen on the adsorption and diffusion of hydrogen. The model reproduces well the experimental data concerning the bistability range, the dependence of the front propagation velocity on the hydrogen partial pressure and temperature, and the parameter-dependent change in front anisotropy. The simulations demonstrate that oxygen diffusion cannot be neglected despite the fact that under typical conditions the rate of oxygen diffusion is several orders-of-magnitude slower than that of hydrogen.