Effortless estimation of basins of attraction

TL;DR

This paper introduces an automated, approximation-free method for identifying attractors and their basins in high-dimensional dynamical systems using a finite state machine approach, applicable to various types of maps and projections.

Contribution

It presents a novel, fully automated technique that works without prior knowledge of attractors and outperforms naive methods in efficiency and accuracy.

Findings

Effective in high-dimensional systems

Handles fractal basin boundaries and chaotic attractors

Open-source implementation available

Abstract

We present a fully automated method that identifies attractors and their basins of attraction without approximations of the dynamics. The method works by defining a finite state machine on top of the system flow. The input to the method is a dynamical system evolution rule and a grid that partitions the state space. No prior knowledge of the number, location, or nature of the attractors is required. The method works for arbitrarily-high-dimensional dynamical systems, both discrete and continuous. It also works for stroboscopic maps, Poincar\'e maps, and projections of high-dimensional dynamics to a lower-dimensional space. The method is accompanied by a performant open-source implementation in the DynamicalSystems.jl library. The performance of the method outclasses the naive approach of evolving initial conditions until convergence to an attractor, even when excluding the task of first identifying the attractors from the comparison. We showcase the power of our implementation on several scenarios, including interlaced chaotic attractors, high-dimensional state spaces, fractal basin boundaries, and interlaced attracting periodic orbits, among others. The output of our method can be straightforwardly used to calculate concepts such as basin stability and final state sensitivity.