

Preprint 83/2007
Entropy, dimension, and state mixing in a class of time-delayed dynamical systems
David Albers and Fatihcan M. Atay
Contact the author: Please use for correspondence this email.
Submission date: 05. Sep. 2007
Bibtex
PACS-Numbers: 05.45.-a, 89.75.-k, 05.45.Tp, 02.30.Ks
Keywords and phrases: chaos, Lyapunov exponents, high dimensions, delay, entropy, Kaplan-Yorke dimension, structural stability
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; and
(iv) the observed phase transition (bifurcation) between
low-dimensional, reducible dynamics
and high or infinite-dimensional dynamics.