Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns

TL;DR

This paper introduces a method to construct a minimal set of embedded unstable recurrent patterns that effectively approximate high-dimensional chaotic attractors, enabling reduced yet accurate representations of complex dynamics.

Contribution

The paper presents a novel approach for creating minimal covers of chaotic attractors using adaptable proximity measures and demonstrates its effectiveness on a high-dimensional system.

Findings

Minimal covers can faithfully approximate chaotic attractors.

Effective reduction of attractor complexity using subspace proximity measures.

Method applicable to high-dimensional spatiotemporal chaos.

Abstract

We propose a general method for constructing a minimal cover of high-dimensional chaotic attractors by embedded unstable recurrent patterns. By minimal cover we mean a subset of available patterns such that the approximation of chaotic dynamics by a minimal cover with a predefined proximity threshold is as good as the approximation by the full available set. The proximity measure, based on the concept of a directed Hausdorff distance, can be chosen with considerable freedom and adapted to the properties of a given chaotic system. In the context of a spatiotemporally chaotic attractor of the Kuramoto--Sivashinsky system on a periodic domain, we demonstrate that the minimal cover can be faithfully constructed even when the proximity measure is defined within a subspace of dimension much smaller than the dimension of space containing the attractor. We discuss how the minimal cover can be used to provide a reduced description of the attractor structure and the dynamics on it.