IEEE Systems, Man and Cybernetics Magazine - October 2021 - 16

An n-dimensional dynamical system has n Lyapunov
exponents, and, if it is chaotic, at least one of them is positive,
resulting in sensitive dependence on the initial conditions.
If at least two Lyapunov exponents are positive, we
speak of hyperchaotic systems. Now, how do we measure
just how chaotic a system is, and how do we compare two
chaotic systems in any way?
One possible entropy-based approach, the Kolmogorov-Sinai
entropy, is related to the positive Lyapunov
exponents and may give us a hint of just how
unpredictable a chaotic system is, with a numerical value
between zero (nonchaotic deterministic systems) and
infinity (purely random systems). The link with Shannon's
entropy and information theory in general
is
straightforward, but interesting nontrivial results linking
Lyapunov exponents of random dynamical systems and
entropy in wireless channels suggest there is more to it
than meets the eye [4].
Another metric is the dimension. In the usual sense,
we perceive the dimension of a geometric construct as
an integer, living in a 3D space, observing images as 2D
projections, and so on. However, if we make a finer measure
of the dimension (e.g., the Hausdorff dimension) to
describe just how much space the object whose dimension
we measure takes, we discover that not everything
is integer dimensional. In particular, observing the
Lorenz attractor in Figure 1 may lead to a conclusion
that it is not exactly 2D: its dimension is, in fact, just
1
0.5
4
2
10
20
30
0.5 10 0.5 10 0.5
20
100
50
1
-1
1
2
1
20
-20
100
0.5
1
2,000
-2,000
-100
01 2
(d)
-20
(e)
20
40
20
40 20
0 -20 -40 20
(f)
10
-10
-20
0.5 10 0.5
1
0.5 10 10
(a)
(b)
20
30
-20
-1 01
(c)
20
Figure 1. The signals in time (top) and corresponding attractors in the phase space (bottom) for (a) aperiodic
convergence, (b) aperiodic divergence, (c) periodic oscillation, (d) oscillatory convergence, (e) oscillatory
divergence, and (f) chaos. The arrows indicate the direction along which the system evolves in the phase space
(from start to end).
16 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE October 2021

IEEE Systems, Man and Cybernetics Magazine - October 2021

Table of Contents for the Digital Edition of IEEE Systems, Man and Cybernetics Magazine - October 2021

Contents
IEEE Systems, Man and Cybernetics Magazine - October 2021 - Cover1
IEEE Systems, Man and Cybernetics Magazine - October 2021 - Cover2
IEEE Systems, Man and Cybernetics Magazine - October 2021 - Contents
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 2
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 3
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 4
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 5
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 6
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 7
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 8
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 9
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 10
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 11
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 12
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 13
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 14
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 15
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 16
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 17
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 18
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 19
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 20
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 21
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 22
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 23
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 24
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 25
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 26
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 27
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 28
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 29
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 30
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 31
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 32
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 33
IEEE Systems, Man and Cybernetics Magazine - October 2021 - 34
IEEE Systems, Man and Cybernetics Magazine - October 2021 - Cover3
IEEE Systems, Man and Cybernetics Magazine - October 2021 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/smc_202310
https://www.nxtbook.com/nxtbooks/ieee/smc_202307
https://www.nxtbook.com/nxtbooks/ieee/smc_202304
https://www.nxtbook.com/nxtbooks/ieee/smc_202301
https://www.nxtbook.com/nxtbooks/ieee/smc_202210
https://www.nxtbook.com/nxtbooks/ieee/smc_202207
https://www.nxtbook.com/nxtbooks/ieee/smc_202204
https://www.nxtbook.com/nxtbooks/ieee/smc_202201
https://www.nxtbook.com/nxtbooks/ieee/smc_202110
https://www.nxtbook.com/nxtbooks/ieee/smc_202107
https://www.nxtbook.com/nxtbooks/ieee/smc_202104
https://www.nxtbook.com/nxtbooks/ieee/smc_202101
https://www.nxtbook.com/nxtbooks/ieee/smc_202010
https://www.nxtbook.com/nxtbooks/ieee/smc_202007
https://www.nxtbook.com/nxtbooks/ieee/smc_202004
https://www.nxtbook.com/nxtbooks/ieee/smc_202001
https://www.nxtbook.com/nxtbooks/ieee/smc_201910
https://www.nxtbook.com/nxtbooks/ieee/smc_201907
https://www.nxtbook.com/nxtbooks/ieee/smc_201904
https://www.nxtbook.com/nxtbooks/ieee/smc_201901
https://www.nxtbook.com/nxtbooks/ieee/smc_201810
https://www.nxtbook.com/nxtbooks/ieee/smc_201807
https://www.nxtbook.com/nxtbooks/ieee/smc_201804
https://www.nxtbook.com/nxtbooks/ieee/smc_201801
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_1017
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0717
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0417
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0117
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_1016
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0716
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0416
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0116
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_1015
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0715
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0415
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0115
https://www.nxtbookmedia.com