  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.
