Adaptive stochastic continuation with a modified lifting procedure applied to complex systems

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models. To analyse the solution and bifurcation structure of such systems on the level of macroscopic observables one has to rely on equation-free methods like stochastic continuation. Here, we investigate how to improve stochastic continuation techniques by adaptively choosing the model parameters. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach to give an analysis of (i) the Ising model in two dimensions, (ii) an active Ising model and (iii) a stochastic Swift-Hohenberg equation. We conclude by discussing the abilities and remaining problems of the technique.