Basin of attraction organization in infinite-dimensional delayed systems: a stochastic basin entropy approach

TL;DR

This paper extends basin entropy to high-dimensional stochastic spaces to analyze the complex, multistable attractor structures in infinite-dimensional delayed systems like the Mackey-Glass model, aiding long-term predictability assessment.

Contribution

It introduces a stochastic basin entropy method for high-dimensional spaces and combines it with basin fraction analysis to study attractor structures in infinite-dimensional delayed systems.

Findings

Quantifies predictability of long-term dynamics.

Characterizes basin structures and intermixing.

Provides tools for analyzing complex infinite-dimensional systems.

Abstract

The Mackey-Glass system is a paradigmatic example of a delayed model whose dynamics is particularly complex due to, among other factors, its multistability involving the coexistence of many periodic and chaotic attractors. The prediction of the long-term dynamics is especially challenging in these systems, where the dimensionality is infinite and initial conditions must be specified as a function in a finite time interval. In this paper we extend the recently proposed basin entropy to randomly sample arbitrarily high-dimensional spaces. By complementing this stochastic approach with the basin fraction of the attractors in the initial conditions space we can understand the structure of the basins of attraction and how they are intermixed. The results reported here allow us to quantify the predictability giving us an idea about the long-term evolution of trajectories as a function of the initial conditions. The tools employed can result very useful in the study of complex systems of infinite dimension.