Untangling Complex Systems, page 75
derives the that thermal gradients in the experiments 1 and 2, i.e., (Δ T) and (Δ
, were
1
T)2
0 0019
.
K < ( T
∆ ) < ( T
∆ ) < 0 0227
.
K. Finally, the height of the solution decreased 0.2 cm
1
2
in 2200 s. Therefore, the power absorbed by the solution was about 0.036 J s−1.
It is possible to experience the “butterfly effect” also by following the instructions of
list B. The spectral evolution for the photochromic SpO under irradiation at 360 nm is
reported in Figure 10.41. Initially, the solution in uncolored. Upon UV, a band centered
at 612 nm forms and the solution becomes colored. The absorbance of the spectral
band into the visible region grows until we reach the photo-stationary state. Then, it
oscillates due to the convective motion of the solvent.
Figure 10.42 reports two spectral trends recorded at 612 nm (graphs a and c), and in two
independent experiments (labeled as 1 and 2, respectively).
1.2
A
0.6
hν
350 400 450 500 550 600 650 700 750
λ (nm)
FIGURE 10.41 Spectral evolution of the photochromic SpO upon UV irradiation.
The Emergence of Chaos in Time
375
1
0.50
0.54
1
A
A
0.45
0.25
0.36
0.00 0 500 1000 1500 2000 2500 3000 3500
1800
2000
2200
(a)
t (sec)
(b)
t (sec)
0.6
2
0.50
2
0.5
A
A
0.25
0.4
0.3
0.00 0 500 1000 1500 2000 2500 3000 3500
1800
2000
2200
(c)
t (sec)
(d)
t (sec)
FIGURE 10.42 Spectral profiles recorded at 612 nm and generated by the “Hydrodynamic Photochemical
Oscillator” based on the photochromic SpO in two distinct experiments, labeled as 1 (graph a) and 2 (graph c).
Graphs b and d show two enlargements of the kinetics 1 and 2, respectively.
Although the experimental conditions were very similar (see the data in Table 10.2, where
the subscripts i and f refer to the initial and final states, respectively), the two profiles of A
versus time are significantly different. It is worthwhile noticing that the oscillations shown in
Figure 10.42 are recordable also at wavelengths where both SpO and MC do not absorb, for
example at 800 nm, confirming that are due to the hydrodynamic motions of the solutions.
The discrepancy between the two kinetics (1 and 2) is further proof of how convection is
sensitive to the initial conditions. In both cases, we have oscillations in the signals of A ver-
sus time, but they are not the same. The Fourier Transforms (see Figure 10.43) demonstrate
that in both cases we have a discrete number of frequencies. However, in experiment 1,
TABLE 10.2
Contour Conditions Relative to the Experiments
1 and 2 Shown in Figure 10.42.
Conditions
Exp. 1
Exp. 2
h
3.1
3.1
i
h
3.0
3.0
f
Ti
25.3°C
25.0°C
Tf
25.4°C
25.2°C
P
967.0 hPa
967.7 hPa
i
P
966.2 hPa
967.0 hPa
f
(Humidity)i
43%
42.6%
(Humidity)f
41.4%
43.7%
376
Untangling Complex Systems
1
0.02
Amplitude
0.000.00
0.03
0.06
0.09
0.12
0.15
Frequency (Hz)
2
0.02
Amplitude
0.000.00
0.03
0.06
0.09
0.12
0.15
Frequency (Hz)
FIGURE 10.43 Fourier spectra for the spectrophotometric kinetics 1 (graph on top) and 2 (graph at the
bottom), respectively.
there is a dominant frequency of 0.0678 s−1, whereas, in experiment 2, more than one fre-
quency has similar amplitudes. For instance, the frequencies of 0.0584 s−1 and 0.0663 s−1.
It is reasonable to assume that in both cases Ra
. By using the values of
c,1 < ( Ra)1or2 < Rac,2
the physical parameters for acetone, we deduce that the vertical thermal gradients in the two
runs are in the range 0 0017
.
K < ( T
∆ ) ≤ ( T
∆ ) < 0 0201
.
K. In both experiments, the volume
1
2
that evaporated was 0.1 cm3 after 3500 s. The power absorbed by the solutions was 0.012 J s−1.
Figure 10.44 shows the hydrodynamic waves spreading MC throughout the solution.
10.15. The data “b1.txt” are plotted in Figure 10.45. The upper graph shows the heart rate, whereas the lower graph reports the trend of chest volume over time. Both the time series
refer to a patient exhibiting sleep apnea, namely a sleep disorder characterized by pauses
in breathing or periods of shallow breathing. The anomalous breathing periods can last
for a few seconds to several minutes.
Before running a routine of TISEAN, you can be acquainted with the necessary inputs
and the right syntax, by typing > name_of_routine -h.
In Figure 10.46, you can see the outputs of the calculations. The graphs in the left col-
umn of the Figure refer to the heart rate time series, whereas the charts in the right column
Time
FIGURE 10.44 Sequence of snapshots showing hydrodynamic convective waves spreading MC throughout
the solution.
The Emergence of Chaos in Time
377
100
ts/min)
(bea 80
t rate ar 60
He
0
2000
4000
6000
8000
t (s)
30000
.u.)
0
Chest volume (a −30000
0
2000
4000
6000
8000
t (s)
FIGURE 10.45 Time series of heart rate (top graph) and chest volume (lower graph) for a patient showing
sleep apnea.
2.1
1.5
1.4
1.0
al information 0.7
al information 0.5
tu
tu
Mu
Mu 0.0
0
5
10
15
20
0
5
10
15
20
(a)
τ
(b)
τ
ors 1.0
ors 1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
2
Fraction of false neighb
4
6
8
10
Fraction of false neighb
2
4
6
8 10 12 14 16 18 20
(c)
m
(d)
m
−2
−2
D
D −4
−4
−6
0
10
20
30
40
50
0
10
20
30
40
50
(e)
Δt
(f)
Δt
FIGURE 10.46 Determination of the time delays (a and b), embedding dimensions (c and d), and Lyapunov
exponents (e and f) for the heart rate and chest volume time series, respectively.
378
Untangling Complex Systems
refer to the chest volume time series. The plots of the first row regard the determination
of the time delay ( τ) through the calculation of mutual information (routine mutual).
It results that the first minimum in (a) is for τ = 6, whereas in (b‖δ(t)‖) is for τ = 2. The
plots of the second row regard the calculations of the fraction of the false nearest neigh-
bors (routine false_nearest). The fraction is 0 for m = 8 in (c) and m = 15 in (d). Therefore, by using the Takens’ theorem, it results that the phase space for the heart rate time series
has τ = 6 as time delay and m = 8 as embedding dimension, whereas that for the chest
volume time series has τ = 2 and m = 15. The largest Lyapunov exponent calculated by
the Kantz’s method (routine lyap_k) for heart rate is λ = 0.011 (graph (e)), whereas that for
the chest volume is λ = 0.033 (graph (f)), indicating the possibility that both time series
are chaotic.
Chaos in Space
11 The Fractals
A fractal is a way of seeing infinity
Benoit Mandelbrot (1924–2010 AD)
11.1 INTRODUCTION
In Chapter 10, we have learned that chaotic dynamics can originate “strange” structures. An exam-
ple is the bifurcation diagram of the logistic map (see Figure 10.9). You probably remember that it describes the evolution of the relative abundance x of a population as a function of its growth rate r.
n
If we zoom in on specific regions of the bifurcation diagram, where “white patches” are interspersed
with clouds of infinite points, copies of the overall diagram reappear at different spatial scales. The
bifurcation diagram is self-similar whatever is the magnification scale. 1
Even the Lorenz model for weather forecasts originated a “strange” attractor in the chaotic
regime (see Figure 10.15). At first sight, the Lorenz attractor looks like a butterfly: it consists of a pair of surfaces, as if it were a pair of wings, merging into the central part—the body of the grace-ful flying insect. If we look at the details of the attractor, we find, just as Lorenz realized, every
chaotic trajectory crosses an infinite number of surfaces and not just two. To gain more insight
into the geometrical structure of the Lorenz’s strange attractor, it is useful to exploit the strategy
contrived by Poincaré while he was studying the three-body problem mentioned in Chapter 10.
Poincaré discovered that the complexity of the analysis of a swirling trajectory can be reduced by
focusing on its intersections with a fixed surface. Such a surface, now known as Poincaré section,
transforms a continuous N-dimensional trajectory into an ( N−1)-dimensional map if the surface
is transverse to the trajectory (see Figure 11.1). The resulting ( N−1)-dimensional map is called Poincaré map.
In particular, the Poincaré map of the Lorenz attractor for z = 27 looks like bundles of sequences
of points (Viswanath 2004). Every bundle of points is self-similar because it maintains its regular
spacing at different levels of magnification (in principle, ad infinitum).
In 1976, the theoretical astronomer, Michel Hénon, after being acquainted with the Lorenz
attractor, formulated a two-dimensional map as a simple model having the essential properties of
the Lorenz system (Hénon 1976). The Hénon map is given by
x
2
1
−
+
n+ = yn
axn 1 [11.1]
yn+1 = bxn [11.2]
where a and b are two adjustable parameters. When a = 1.4 and b = 0.3, the Hénon map gives rise to a strange attractor, looking like a boomerang and shown in Figure 11.2.
Successive enlargements of the exterior border of the boomerang-like shape reveal the fine
structure of the attractor. In the first picture, the border consists of three lines. By zooming into the
1 The only limit that exists is imposed by the degree of accuracy of our calculations. In other words, the limit depends on the number of significant figures we used in building the bifurcation diagram of the logistic map.
379
380
Untangling Complex Systems
rk+2
rk+1
rk
FIGURE 11.1 Representation of a Poincaré section that transforms a continuous three-dimensional trajec-
tory into a bi-dimensional map.
0.4
Hénon attractor
0.195
Z1
0.3
0.19
0.2
Z1
0.185
0.1
0.18
Z2
y
0
0.175
0.17
−0.1
0.165
−0.2
0.16
−0.3
0.155
−0.4−1.5
−1
−0.5
0
0.5
1
1.5
0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82
0.1778
Z3
0.181
Z2
0.18
0.1776
0.179
0.1774
0.178
Z3
0.1772
y
yy 0.177
0.177
0.176
0.1768
0.175
0.174
0.1766
0.173
0.1764
0.704 0.705 0.706 0.707 0.708 0.709 0.71 0.711 0.712 0.713 0.714
0.685 0.69 0.695 0.7 0.705 0.71 0.715 0.72 0.725 0.73 0.735
x
x
FIGURE 11.2 A series of three enlargements (Z1, Z2, and Z3) into a portion of the exterior border of the
Hénon attractor.
rectangle Z1 of Figure 11.2, we see that there are six lines: a single curve, then two closely spaced curves above it, and then three more. An enlargement in Z2 reveals that the three lines are actually
six curves; grouped one, two, three, precisely as in Z1. A further expansion of the last three lines
unveils the structure also observed in Z1; i.e., one, two and three lines. And so on. Self-similarity
can continue indefinitely. The limit is imposed only by the number of decimal figures we use in
our calculations.
Chaos in Space
381
TRY EXERCISE 11.1
Self-similarity can be encountered in every strange attractor. In fact, the strange attractors are
examples of FRACTALS.
11.2 WHAT IS A FRACTAL?
A fractal is a complex geometric shape. It maintains its sophisticated form, or the main features of
its complex structure, under different levels of magnification, i.e., under changes of spatial scale.
This peculiarity of fractals is defined as either “scale invariance” or “scale symmetry,” or more
often, “self-similarity.” Several fractals can be built by following simple rules. A fractal that often
appears in the strange attractors generated by chaotic dissipative dynamics is the Cantor set. For
example, it is the fractal that can be detected in the Lorenz attractor. The set is named after the
nineteenth-century mathematician Georg Cantor. However, it was created by Henry Smith, who
was a nineteenth-century geometry professor at the Oxford University. 2 To build a Cantor set, we start with a bounded interval C . We remove an open interval from inside . is divided into two
0
C 0 C 0
subintervals, each containing more than one point. Iteratively, we remove an open interval from
each remaining interval. In this way, the length of the remaining intervals at each step shrinks to
zero as the construction of the set proceeds. After an infinite number of iterations, we obtain the
Cantor set. The most often mentioned Cantor set is that obtained by removing the middle third of
every interval (see Figure 11.3). In the end, it consists of an infinite number of infinitesimal pieces, separated by gaps of various lengths, and having a negligible length.3 Of course, it is impossible to print a fractal on a piece of paper.
TRY EXERCISE 11.2
Another well-known and perfectly self-similar fractal is the Koch curve, contrived by the Swedish
mathematician Helge von Koch. For its construction, we start with a horizontal segment ( K ). Then,
0
we delete the middle third of K , and we replace it with the other two sides of an equilateral triangle,
0
and we obtain K (see Figure 11.4). In the subsequent stages, we repeat recursively the same rule: 1
C 0
C 1
C 2
C 3
C 4
FIGURE 11.3 Scheme showing the first four steps for constructing the Cantor set by removing the middle
third of every interval. It has been obtained by using the “Examples of Fractals” model, Complexity Explorer
project, http://complexityexplorer.org.
2 The first self-similar fractals were invented earlier than the discovery of strange attractors. In the beginning, they were merely “pathological phenomena” interesting only mathematicians.
3 Cross sections of strange attractors are often Cantor set devoid of exact self-similarity. However, they are topological Cantor sets. A topological Cantor set C has two properties. First, C contains only single points and no intervals. Second, every point in C has a neighbor arbitrarily close by. A famous example of topological Cantor set is the logistic map
