IEEE Systems, Man and Cybernetics Magazine - October 2021 - 14

We present a short overview of dynamical systems
and chaos before focusing on the applications of dynamical
systems theory to wireless communications in the
past 30 years, ranging from modeling on the physical
layer to the different kinds of self-similar traffic encountered
all the way up to the network layer. The examples of
past research and its implications are grouped and
mapped onto the media layers of the International Organization
for Standardization (ISO) open systems interconnection
(OSI) model to show just how ubiquitous
dynamical systems theory can be and trace the paths
that may be taken now.
When considering future paths, we argue that the time
has come for us to revive interest in dynamical systems for
wireless communications. It did
not happen already because of the
big question: can we afford to consider
systems of interest as dynamical
systems, and what are the
tradeoffs? The answers to this
question are dynamical systems of
their own: they change not only
with the modeling context, but
also with time.
In the present moment, available
resources allow such an
approach, and the current demands
ask for it. Reservoir computing, the
major player in dynamical systemsrelated
learning, originated in wireless
communications, and to wireless communications it
should return.
time of the throw makes it look random. (The level of
understanding of the processes and granularity of the
model ties closely with the traditional understanding of
uncertainty in engineering models [1], [2].)
While randomness is usually portrayed as the opposite
of determinism and modeled in a different fashion, dynamical
systems embrace it as much as they embrace determinism.
Random dynamical systems are a generalized
version of deterministic ones, as they allow for a stochastic
component. In wireless communications, this is the
channel equation we start with: a linear random dynamical
system with noise as a random component.
The history of dynamical systems theory shows its
When considering
future paths, we
argue that the time
has come for us
to revive interest
in dynamical
systems for wireless
communications.
early bond with wireless communications, as some of the
fundamental dynamical systems
theory concepts introduced by
Poincaré came from his wireless
telegraphy seminars (1908). The
development of both disciplines in
the following decades continued,
going hand in hand with the oscillators
that were the central object
of interest in dynamical systems
theory and an invaluable component
of every radio device from
their inception.
The perspectives we investigate
here are those opened in dynamical
systems theory once the theory
of deterministic chaos was estabHistory
of Dynamical Systems
Everything is a dynamical system, depending on how you
define everything (and how you define a dynamical system).
In the realm of wireless communications, we can
observe quantifiable system outputs evolving in time (an
essential property of a dynamical system) at every level of
the ISO model hierarchy, where some aspects of the behavior
are seen as governed by simple rules and very ordered,
while others are seemingly random. The random cases
attract more attention, as they are more difficult to predict
and harness, and they usually include more interaction
with the external factors: users and the environment. The
question of whether to model these parts as dynamical
systems is a multilayered one.
Not all systems can be described by a low-order, lowcomplexity
model. We could model the universe using elementary
particle positions and momenta at a certain point
in time, but the computational power for such a variant of
Laplace's demon surpasses the information limits of the
same universe greatly. Measurements might be hard to
obtain as well, and, while throwing a die is a perfectly
deterministic process with simple dynamics, not knowing
the precise (and ever-changing) initial conditions at the
14 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE October 2021
lished in the second half of the 20th century, with the
notions of sensitive dependence on the initial conditions,
fractal dimension attractors, ergodicity, and so on. The
path we chose is one of understanding the already existing
dynamical phenomena within wireless systems and putting
them to use.
The other road in observing dynamical systems in wireless
communications is the one of chaotic communications.
This has been the dominant topic in the area since
the early 1990s, when the possibility of the synchronization
of two chaotic systems was demonstrated [3]. This
topic has repeatedly been surveyed in the past, and it represents
the chaos added to a communication system, not
the one found existing within it.
We begin our story by presenting the basics of dynamical
systems theory. This will help us appreciate the
efforts made in the past to identify elements of dynamical
systems in wireless communications settings. These
efforts are then presented systematically, mapped onto
the media layers of the ISO OSI model to put the concepts
into a context and suggest ways to proceed with
research today.
Then, we proceed with our central claim: the combination
of dynamical systems theory and machine learning
has the potential to radically change wireless communications
performance. We offer some initial results motivated

IEEE Systems, Man and Cybernetics Magazine - October 2021

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