IEEE Systems, Man and Cybernetics Magazine - October 2021 - 17

slightly larger than two. The noninteger dimension of
strange attractors is thus another indicator and a measure
of chaos.
When speaking of noninteger dimensions, it is necessary
to mention self-similarity and fractals as one of the
most often mentioned features in dynamical systems and
chaos, at least in the popular view. Fractals are self-similar
structures, as a zoomed in part looks just like a bigger
one, iterating the structure. The infinitely rough structure
of a fractal results in its noninteger Hausdorff
dimension while retaining the integer topological dimension
(e.g., fractals with the topological dimension of two
can fit in a plane).
Up and Down the Media Layers
The 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. The development
of the algorithms described in the previous section
to determine the chaoticity of the data (and the systems
generating it) brought a series of investigations and
results in different areas of science and engineering.
Chaos was looked for in the phenomena previously considered
random (from the stock market to temperature
oscillations), and dynamical system formulation was
sought in the fields where the systems were not considered
quantitative at all, such as learning processes and
interpersonal interaction.
Wireless communications followed suit, and the
aspects examined were all over the media layers of the
ISO OSI model describing it. The logic behind this is simple:
where there is a time series or signal to be measured,
there is a dynamical system producing it. As we have
seen, we can find out the details about the system from
the outputs, even more so if it is chaotic. A preview of this
historical review is given in Figure 3 and elaborated upon
in this section. Illustrative references related to different
layers are presented in Table 1.
Physics
The wireless channel has been viewed as a dynamical system
from the early beginning, but it keeps a lot of its effect
Black Box
Inaccessible Outputs/States
Accessible Outputs
The Signal
Processing Algorithms
Automutual
Information
x
The Attractor
t
Cao Algorithm
Signal Metrics
Parameters:
Lag and Dimension
Reconstruction
and Embedding
System Metrics
Hurst Exponent
Lyapunov Exponent
Figure 2. The tool chain for dynamical systems and chaos. The attractor reconstruction is based on the Takens
theorem, which shows that it is possible to reconstruct an attractor based on a single output signal and
delayed versions of it serving as the other state variables. The delay (lag) that should be used for the remaining
state variables can be determined based on the automutual information function of the signal. The use of
the automutual information function is another hint of how closely intertwined dynamical systems theory and
information theory are. After determining the delay and applying an algorithm to determine the dimension, the
attractor is going to be embedded; i.e., in the order of the system, an attractor can be generated. While not
perfectly the same as the original, it can reveal a lot about s system's dynamics and serve as a foundation for
the calculations of relevant metrics. The importance of this process is the transition from signals to systems, from
the particular inputs and outputs to the general mechanism.
October 2021 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 17

IEEE Systems, Man and Cybernetics Magazine - October 2021

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