Monte Carlo Basin Bifurcation Analysis

TL;DR

This paper introduces a machine learning-based numerical method for analyzing complex high-dimensional systems by characterizing their asymptotic states and basins of attraction through clustering trajectory statistics.

Contribution

It presents a novel, flexible approach combining random sampling and clustering to analyze complex systems, applicable beyond traditional bifurcation analysis.

Findings

Successfully applied to oscillator networks and agent-based models

Effectively identifies multiple asymptotic states and basins of attraction

Provides a modular Julia package for broad applications

Abstract

Many high-dimensional complex systems exhibit an enormously complex landscape of possible asymptotic states. Here, we present a numerical approach geared towards analyzing such systems. It is situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis. With our machine learning method, based on random sampling and clustering methods, we are able to characterize the different asymptotic states or classes thereof and even their basins of attraction. In order to do this, suitable, easy to compute, statistics of trajectories with randomly generated initial conditions and parameters are clustered by an algorithm such as DBSCAN. Due to its modular and flexible nature, our method has a wide range of possible applications. Typical applications are oscillator networks, but it is not limited only to ordinary differential equation systems, every complex system yielding trajectories, such as maps or agent-based models, can be analyzed, as we show by applying it the Dodds-Watts model, a generalized SIRS-model. A second order Kuramoto model and a Stuart-Landau oscillator network, each exhibiting a complex multistable regime, are shown as well. The method is available to use as a package for the Julia language.