Entropy, dimension, and state mixing in a class of time-delayed dynamical systems

TL;DR

This paper investigates how adding time-delay coordinates affects the complexity and entropy of dynamical systems, revealing phase transitions, high-dimensional behaviors, and spectral properties through analytical and computational methods.

Contribution

It provides new analytical insights into entropy invariance, explores the effects of past state mixing, and identifies phase transitions in high-dimensional delay systems.

Findings

Entropy remains invariant under simple time re-scalings.

High-dimensional, high-entropy dynamics emerge from non-trivial past state mixing.

A phase transition between low-dimensional and high-dimensional dynamics is observed.

Abstract

Time-delay systems are, in many ways, a natural set of dynamical systems for natural scientists to study because they form an interface between abstract mathematics and data. However, they are complicated because past states must be sensibly incorporated into the dynamical system. The primary goal of this paper is to begin to isolate and understand the effects of adding time-delay coordinates to a dynamical system. The key results include (i) an analytical understanding regarding extreme points of a time-delay dynamical system framework including an invariance of entropy and the variance of the Kaplan-Yorke formula with simple time re-scalings; (ii) computational results from a time-delay mapping that forms a path between dynamical systems dependent upon the most distant and the most recent past; (iii) the observation that non-trivial mixing of past states can lead to high-dimensional, high-entropy dynamics that are not easily reduced to low-dimensional dynamical systems; (iv) the observed phase transition (bifurcation) between low-dimensional, reducible dynamics and high or infinite-dimensional dynamics; and (v) a convergent scaling of the distribution of Lyapunov exponents, suggesting that the infinite limit of delay coordinates in systems such are the ones we study will result in a continuous or (dense) point spectrum.