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

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 (UPOs) 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, composed of time-shifted copies of the data. We then examine the influence of the width and height of the Hankel matrix on the geometry of UPOs 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 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 UPOs in the embedded space. In addition, we present a modified formula to derive symbolic representation of distinct periodic orbits for a specified sequence length, extending the Redfield–Pólya (R–P) enumeration theorem.