IEEE Systems, Man and Cybernetics Magazine - October 2021 - 18

on the signal under the umbrella of random noise and
unpredictable changes. Chaos theory tools and new trends
in dynamical systems theory have provided some hope of
distinguishing the magic of noise from the science of complexity
and chaos in the channel.
Detecting low-order chaos in multipath propagation in
the early 1990s [5] opened some promising paths for the
future, but similar experiments performed later showed
that the deterministic chaos is not always present. The
focus turned to sea clutter in a radar context, but, again,
the results have been inconclusive, coming short of the
revolutionary applications chaotic sea clutter nature might
have in the detection and tracking of targets near the sea
surface, if and when confirmed [12].
For a moment, examine the billiard ball system in which
a tiny difference in the initial condition drastically changes
the trajectory of a billiard ball, depending on the table shape
and position of the other balls. The billiard ball analogy
works, to an extent, in the context of electromagnetic
waves in a chamber and explains ray chaos: the divergence
of two electromagnetic waves originating from almost the
same place due to multiple bounces off the environment [6].
Ray chaos is achievable in other wave settings as well,
e.g., the acoustic case. The case of sound waves is actually
the one from which we borrow a useful application of ray
chaos for radio: enabling good time reversal.
Time reversal in acoustics is the idea of creating a
sound wave that is a time-reversed version of a given
sound wave received: like an echo, but converging into the
point of the original wave's source. This idea [13] extends
to optics and radio communications, serving as the core
principle of several localization and communication
schemes, e.g., conjugate beamforming.
In a rich scattering environment with a lot of reflections,
the bounces in the multipath quickly map the entire
space and allow good time reversal. A chaotic environment
has this property as well, as it exhibits ergodicity: the
system will eventually pass through any part of its attractor,
if we can afford to wait.
Time in dynamical systems can be either a discrete or a
continuous variable. An interesting example of blurring
the lines between the two is the calculation of a continuous-time
limit of a discrete system. The continuous model
is a limit of a discrete model as the discrete time interval
shrinks. If the systems are spatial instead of temporal (i.e.,
a distance plays the role of time), the shrinking distance
produces a continuum in the limit process.
In a network, the infinitely increasing density aims for
the limiting continuous process [7] (Table 1). The resulting
continuous system may either be an approximation that is
good enough or a way to obtain the lower/upper bounds on
the discrete system performance.
Data Link Layer
Taking a step up on the ladder, we stay in the realm of
channel modeling. The question of channel errors, their
Control of a Dynamical System Over
Wireless: Questions of Delay,
Feedback, Priority, Reliability, and Inertia
Is the wireless
traffic self-similar?
Network Layer
Can a combination of chaotic
attractors model channel errors?
Media access control traffic can be modeled as a
continuous (fluid) variable, while the state of nodes
remains Boolean.
Data Link Layer
Self-Similarity and its Effects in
Antenna Design: Fractal Antennas
Ray Chaos: Chaotic Radio Signal Trajectories
in a Suitably Shaped Cavity
Physical Layer
Cross Layer
Figure 3. What has been done in dynamical systems research over the years? Fractals, chaos, and dynamical
systems control have been put to use on different layers of the media part of the ISO OSI model.
18 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE October 2021
Brain-Inspired
and -Directed
Communications
Control of the Wireless Dynamical
System: Questions of Resource
Allocation, Antenna Selection,
Beamforming, Power Control,
and Feedback

IEEE Systems, Man and Cybernetics Magazine - October 2021

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