Separation of periodic orbits in the delay embedded space of chaotic attractors

TL;DR

This paper investigates how time-delay embeddings can be used to distinguish and analyze unstable periodic orbits in chaotic attractors, revealing clear separation of orbits in the embedded space and extending symbolic dynamics methods.

Contribution

It introduces a framework for separating periodic orbits in delay-embedded space and extends symbolic orbit enumeration techniques for better analysis.

Findings

Increasing delay length enhances orbit separation in the embedded space.

Hankel matrix singular vectors form a basis for embedding periodic orbits.

Modified symbolic formula extends Polyá-Redfield enumeration for specific sequences.

Abstract

This work explores the intersection of time-delay embeddings, periodic orbit theory, and symbolic dynamics. Time-delay embeddings have been effectively applied to chaotic time series data, offering a principled method to reconstruct relevant information of the full attractor from partial time series observations. In this study, we investigate the structure of the unstable periodic orbits of an attractor using time-delay embeddings. First, we embed time-series data from a periodic orbit into a higher-dimensional space through the construction of a Hankel matrix, formed by arranging time-shifted copies of the data. We then examine the influence of the width and height of the Hankel matrix on the geometry of unstable periodic orbits in the delay-embedded space. The right singular vectors of the Hankel matrix provide a basis for embedding the periodic orbits. We observe that increasing the length of the delay (e.g., the height of the Hankel matrix) leads to a clear separation of the periodic orbits into distinct clusters within the embedded space. Our analysis characterizes these separated clusters and provides a mathematical framework to determine the relative position of individual unstable periodic orbits in the embedded space. Additionally, we present a modified formula to derive the symbolic representation of distinct periodic orbits for a specified sequence length, extending the Poly\'a-Redfield enumeration theorem.