IEEE Systems, Man and Cybernetics Magazine - October 2021 - 15

by recent developments in the field as a motivation for further
work.
Dynamical Systems and Chaos
While chaotic behavior remains the trademark of
dynamical systems theory and the most interesting
exhibit in its zoo, it also is still a rare catch. When we
aim to understand a dynamical system, we are interested
in its stability, periodicity, controllability, and observability:
the properties of the system acting on its own
and under our influence.
Giving attributes of a dynamical system to signal components
previously considered to
be random noise 1) allows a better
prediction, which, in turn,
enables the better suppression of
interference; 2) opens an opportunity
to examine it as " a feature,
not a bug " -i.e., adds another
degree of freedom; and 3) offers
a physical interpretation.
Signals, Phase Space,
and Attractors
A dynamical system is, once we
know all of its degrees of freedom
and sources of dynamics, a system
of differential or difference equations,
depending on whether we
work in continuous or discrete
time. However, we tend to know so much only about very
simple models seen in nature or those we devise ourselves.
Usually, a dynamical system seen in the wild is a black box
for us.
Both a system of equations and a black box take inputs,
However, chaotic systems were found not to follow any
of these patterns: they end up confined in a bounded part
of state space called the attractor (unlike unstable systems),
traverse it in a nonperiodic manner (unlike periodic
systems), and never converge to a single equilibrium
(unlike stable systems). Often, it is an example of motion
around two equilibria and jumping from orbiting one to
the other, as seen in the celebrated Lorenz attractor
shown in Figure 1(f).
The evolution of trajectories on the attractor demonThe
quest for chaos
and other interesting
dynamical system
properties was a
hot topic with all
of the features of
a bandwagon at
the end of the last
century.
strates two important properties of a chaotic system. The
first one is the sensitive dependence on initial conditions,
as two arbitrarily close phase
space trajectories will separate
exponentially fast on the attractor,
rendering the prediction of future
motion impossible in the long run.
The second one is ergodicity: a
phase trajectory will get arbitrarily
close to any point on the attractor,
given enough time has passed.
The signals and attractors are
easily obtained when the mathematical
model of the system exists:
even though the nonlinear differential/difference
equations governing
the dynamics are usually not solvable
in closed form, a numerical
solution can be found. However,
while the system is a black box, we
usually have only a few (typically, only one) system outputs
available.
How do we reconstruct the other state variables? How
change their states, and produce outputs, all of which
change in time. The number of state variables of a system
is its order, the order of the equations' system in case we
have a mathematical description. The outputs are usually
some of the states of the system visible to us.
Traditionally, system identification, i.e., building a
model from the limited knowledge about the black box, is
a matter of statistics and special sets of test inputs. This is
the way the wireless channel is estimated with a (pilot)
signal. Often, we try to obtain static or linear dynamic
models, as they are easy to work with. However, they are
usually valid only within a narrow time or parameter interval.
Nonlinear systems ask for different identification and
modeling methods.
Having n state variables, it is often useful to plot them in
an n-dimensional coordinate system, the phase space. This
is where phase trajectories are observed; typical examples
are shown in Figure 1. Unstable systems may diverge to
infinity either quasi-periodically or aperiodically; stable systems
may converge to an equilibrium in the same manner,
while periodical systems remain confined to a cycle.
many should be reconstructed in the first place? The signal
analysis and processing for chaotic dynamical systems
has a toolbox for this task. While a detailed description
goes out of the scope of this article, Figure 2 gives an overview
of the attractor reconstruction and quantitative analysis
of the results.
Metrics
Dynamical systems are quantitative, so many metrics
are devised to assess and categorize them. From
the viewpoint of control and stability, measures of
stability margins describe how stable and robust a
system is and how much disturbance it could take
without a failure.
For chaotic systems, the measure called the Lyapunov
exponent gained importance. In a conventional, nonchaotic
deterministic system, it is a negative exponent that
describes how fast two separate phase trajectories converge.
In chaotic systems, however, we have already learned
that even infinitesimally close trajectories diverge at an
exponential rate: a positive Lyapunov exponent describes
this dynamic and, thus, became a symbol of chaos and a
basis for its measure.
October 2021 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 15
IEEE Systems, Man and Cybernetics Magazine - October 2021

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