Untangling Complex Systems, page 79
ranking of cities by size, the distribution of income within an economic system, the relative
growth of staff within industrial firms, the fluctuations in stock prices, and the social differ-
entiation and division of labor in primitive societies. Power laws recur even in geology. The
Gutenberg-Richter law (Gutenberg and Richter 1956), describing the link between earthquake
magnitude ( Ma)17 and its frequency ( f) is
f
a b Ma
=
− (
)
10
[11.30]
where a and b are two constants. Omori’s law (Utsu et al. 1995) that predicts the frequency of after-
shocks ( ν) is a power law of time ( t):
k
ν =
[11.31]
c + t p
(
)
with c and p being constants.
17 The magnitude in the Richter scale is the base-10 logarithmic scale of the ratio ( R) of the amplitude of the seismic waves to an arbitrary minor amplitude, as recorded on a standardized seismograph at a standard distance. Since Ma = log10( R), equation [11.30] is a power law: f
a R b
=
−
10 (
).
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Untangling Complex Systems
Y
Y
(a)
2.0×108 4.0×108 6.0×108 8.0×108 1.0×109 (b)
2.0×106 4.0×106 6.0×106 8.0×106 1.0×107
X
X
Y
Y
(c)
2.0×104 4.0×104 6.0×104 8.0×104 1.0×105
0
250
500
750
1000
X
(d)
X
FIGURE 11.20 Self-similarity of a power law. The graphs (a)–(d) zoom in on the power law.
The recurrence of power laws is reminiscent of fractal phenomena. A power law is itself self-
similar (Mitchell 2009) like any fractal. Figure 11.20 demonstrates that the power law is scale-
invariant. It has the same shape, whatever is the scale of the plot.
An effective strategy to recognize power laws is to build log-log plots of experimental
data. In fact, a true power law appears as a straight line in a double logarithmic axis plot (see
Figure 11.21). Most of the natural, social, and economic phenomena, cited earlier, when ana-
lyzed in a sufficiently wide range of possible values of their variables, may exhibit a limited
linear regime followed by a significant curvature. Such trend is well-fitted by a stretched expo-
nential function of the type
x c
−
x 0
y = Ae
[11.32]
where 0 < c < 1 and x are constant parameters (Figure 11.21). When c =
0
1, the stretched exponential
function transforms in a pure exponential function.
The fact that most of the natural, social and economic distributions display a curved trend in a
double logarithmic axis plot, decaying faster and leading to thinner tails than a power law, has been
interpreted in terms of finite-size effects (Laherrère and Sornette 1998). The natural shapes and
phenomena look like fractals, but they are not genuine fractals in the mathematical sense because
this would mean a self-similarity ad infinitum.
Chaos in Space
401
Power ( A = 1, b = 0.3)
10
Stretched ( A = 1, x 0 = 1, c = 0.3)
Exponential ( A = 1, x
1
0 = 1, c = 1)
0.1
0.01
1E-3
Y 1E-4
1E-5
1E-6
1E-7
1E-80.1
1
10
100
1000
X
FIGURE 11.21 Representation of a power law with its exponent b = 0.3 (continuous line); a stretched exponential with c = 0.3, x 0= 1(dashed line), and an exponential function (dashed-dotted line).
11.13 WHY DOES CHAOS GENERATE FRACTALS?
At the beginning of this chapter, we learned that chaotic systems manufacture fractals. For this rea-
son, fractals are also named as “Chaos in Space.” How is it possible that the “Chaos in Time” gives
rise to fractals? Chaos appears only in the nonlinear regime. Whereas the linearity means “stretch-
ing,” the nonlinearity implies “stretching” and “folding.” In the nonlinear regime, the continual rep-
etition of the two actions, “stretching” and “folding,” generates fractals. Let us consider a chaotic
system in its phase space. When we study its time evolution, we start from a restricted portion of the
phase space, which includes all the possible initial conditions that are indistinguishable (due to the
unavoidable uncertainty in defining the initial conditions). This portion of the phase space could be
a sphere or a cube or any other simple volume.18 If we let the system evolve, each point traces its trajectory. The portion of the phase space embedding all the initial points changes its shape during the
evolution of the dynamics. The countless repetition of “stretching” and “folding” actions transforms
the original shape into a fractal. And the metamorphosis is complete at the infinite time. We can
make this idea appetizing and easy to remind if we think about a pastry chef making a croissant. The
chef maneuvers the dough patiently. The dough represents the portion of the phase space embedding
all the possible indistinguishable initial conditions. The chef stretches the dough with a rolling-pin.
Then, he puts a layer of butter on the dough and folds it. He stretches it again, puts another layer of
butter and folds it again. These actions are repeated many times. Ideally, ad infinitum. Finally, after
cooking the dough, the chef gets a croissant. And a croissant is a fractal-like structure. It is evident
if we cut the croissant into two halves, and we observe them in the direction parallel to the table (see
Figure 11.22). Both halves of the croissant show an almost infinite number of layers.
This culinary similarity is also suitable to explain the sensitivity to the initial conditions of the
chaotic dynamics. In fact, if we consider two close points in the initial spherical dough (representing
two distinct initial conditions), then, just after the first stretching, they are pulled apart. The folding
can approach them. But, when the chef stretches the dough again, they even get farther apart. In one
of the many folding operations, the two points may end up in different layers. Since then, the two
points have two completely different evolutions, likewise the two trajectories in their phase space.
18 When the dimensions of the phase space are larger than three, the portion of the phase space including the initial conditions will be either a hypersphere or a hypercube or any other simple hypervolume.
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Untangling Complex Systems
…
Initial
condition
Stretching
Folding
Stretching
Folding
Stretching
Final
output
FIGURE 11.22 A fractal-like croissant is obtained by stretching and folding the dough, many times.
11.14 CHAOS, FRACTALS, AND ENTROPY
We know very well that any “dissipative” system abides by the Second Law of Thermodynamics and
increases entropy because, in the course of its evolution, it degrades potential and/or kinetic energy into
heat. We also know that any “conservative” non-chaotic system does not produce entropy because the
sum of its kinetic and potential energies remains constant when it evolves. Non-chaotic Hamiltonian
systems are theaters of reversible transformations. But, what about the entropy change in the evolution
of a chaotic “conservative” system (Baranger 2000)? As we already know very well, the initial condi-
tion of the system can only be defined with a finite degree of accuracy. All the equally likely initial
points are confined within a portion of the phase space whose volume is V
Ni
i = ∑
n , where N is the
=1 k
k
i
total number of points contained in V . The initial uncertainty19 we have about the state of the system is i
Ni
Hi = −
pk log2 pk = log2 V
( i
∑
) [11.33]
k=1
If we now let the system evolve, the shape of the volume containing all the initial points changes
drastically. In fact, each point inside V follows its own trajectory. Points that are nearby at the
i
beginning, then, diverge away from each other in an exponential manner. Due to the constant action
of “stretching” and “folding” events, tendrils appear and knots of all sorts form. The structure
becomes finer and finer until we get a fractal (see Figure 11.23).
In the Hamiltonian mechanics, an important theorem holds. It is the Liouville’s theorem stating
that the initial phase space volume, defined by the sum of all possible initial conditions, does not
change. In other words, the final volume of the phase space, Vf , including the trajectories of all the
i
f
Time
FIGURE 11.23 Metamorphosis of a spherical portion of the phase space for a chaotic Hamiltonian system.
19 Remember the Theory of Information formulated by Claude Shannon and presented in Chapter 2.
Chaos in Space
403
FIGURE 11.24 The smooth black bag circumscribes the fractal generated by the chaotic Hamiltonian
system.
initial points, contained in Vi, is equal to the initial volume: Vf = Vi. This condition would mean that the uncertainty at the end of the transformation, H
N f
f = −∑
p log2
log ( ), would be equal to H
=1 k
pk =
2 Vf
k
i.
However, the final shape is a fractal, full of holes, knots, and tendrils. It is a complex structure (see
Figure 11.23)! For the determination of its volume, we are forced to circumscribe it within a larger, smooth bag, like in Figure 11.24.
The outcome is that the final volume V ′ f > Vi. Therefore, the entropy variation given by
∆ S = Hf − Hi is positive. The chaotic dynamics gives rise to a fractal, which is “Chaos in Space.”
The fractal-genesis brings about an increase of our ignorance and hence entropy.
11.15 KEY QUESTIONS
• What is the common feature of the strange attractors?
• What is a fractal?
• How do we determine the dimension of a self-similar fractal?
• How do we obtain the Mandelbrot and the Julia sets?
• Can we describe the “roughness” of many natural complex shapes through the fractal
geometry? Are there genuine fractals in nature?
• How can you determine the dimension of a fractal-like structure?
• Describe the factors that contribute to the formation of patterns in a Hele-Shaw cell.
• Describe the properties of dendritic fractals.
• What is a multifractal?
• What characterizes the diffusion in a fractal?
• What does mean fractal kinetics?
• What is the difference between power laws and stretched exponential functions?
• Why does chaotic dynamics generate fractals?
• Can a Hamiltonian system produce entropy?
11.16 KEY WORDS
Strange attractors; Self-similarity; Dimension of an object; Dendrites; Multifractals; Fractons;
Power Law; Stretched Exponential Function.
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Untangling Complex Systems
11.17 HINTS FOR FURTHER READING
Three other introductory books regarding fractals are that by Feldman (2012) and the two volumes
by Peitgen et al. (1992a, 1992b).
11.18 EXERCISES
11.1. Prove that the Hénon attractor is self-similar if you expand the exterior border of the
boomerang-like structure after computing at least one million successive iterates of equa-
tions [11.1 and 11.2], starting from the origin (i.e., [ x , ]
0 y 0 = [0, 0]).
11.2. The first steps needed to build a Cantor set by eliminating the middle third of every inter-
val are depicted in Figure 11.3. Assume that the length of C is 1 in arbitrary units (
0
a.u. ).
Which is the length of the two segments of C ? What is the length of each line segment
1
in C ? And total length? Try to answer the same questions for . Then, determine the
2
C 2
C 3
general expression defining the number of line segments, the length of each line segment,
and the total length at the n-th iteration. After an infinite number of iterations, which is
the number of line segments, and which is the full length?
11.3. The first steps needed to build a Koch curve are shown in Figure 11.4. Assume that the length of K is 1 in arbitrary units (
? What is the
0
a.u. ). Which is the total length of K 1
length of each line segment in K ? And entire length? Try to answer the same ques-
2
K 2
tions for K . Then, determine the general expression defining the number of line seg-
3
ments, the length of each line segment, and the total length at the n-th iteration. After
an infinite number of iterations, which is the number of line segments and which is the
full length?
11.4. Consider the Sierpinski gasket shown in Figure 11.8. Find the answers to the following questions in iterations 1, 2, 3, and n → ∞:
1. What are the number of triangles?
2. If A is the total area of the triangle at S , which is the area of each triangle and the full
0
area of the structure?
3. If L is the length of one side in S , which is the perimeter of each triangle and the total
0
perimeter of the figure?
11.5. A Cantor set different from that depicted in Figure 11.3 can be obtained by dividing the initial segment into five equal pieces, delete the second and the fourth subintervals, and,
then, repeat this process, indefinitely. Determine the dimension of this fractal, called the
even-fifths Cantor set since the even fifths are removed at each iteration.
11.6. If a square is scaled up by a factor of 3, what happens to its area? If the Sierpinski gasket
is scaled up by a factor of 3, what happens to its size?
11.7. Construct the Lévy C curve fractal by using the following instructions: start from a seg-
ment (LC ). Use the segment as the hypotenuse of an isosceles triangle with angles of 45
0
°,
90°, and 45°. Then, remove the original segment, and you obtain LC . At the second stage,
1
repeat the procedure for the two segments, conceiving both of them as the hypotenuses
of two equivalent isosceles triangles. Repeat the steps many times. How does the fractal
look like? Which is its dimension?
11.8. Solve iteratively equations [11.7] by starting by the following seeds: ( z ,
0 c) = (0, + i);
(0, − i); (0, 2 − i); (0, 0.01 − 0.5 i). Calculate how the distance r of z = a + ib from the origin of the graph, changes after each iteration by using the formula r = a 2 + b 2 .
11.9. Calculate the fate of the seeds z 0= 0, 0.5, 1 for c = 0 and 1.
11.10. Do the seeds (− .
0 12 + .
0 75 i) and (0.4 − .
0 3 i) belong to the Mandelbrot set? In other words,
do their distances decrease after some iterations of equation [11.7]?
11.11. The universe is another example of a natural fractal. Which fractal set does the universe
look like?
Chaos in Space
405
11.12. Print the map of Norway on an A4 paper and determine the dimension of its border by
using the Box Counting method. For the application of the Box Counting method, pre-
pare four grids, based on unit-squares having sides 2.5, 2, 1.5, 1 cm long, respectively,
and print them on transparencies.
11.13. Build your Hele-Shaw cell. By using both pure glycerin and 4% PVA cross-linked by
addition of sodium borate (for example, add 0.5 mL of 4% Na B O to 25 mL of 4
2 4
7
%
PVA) as viscous fluids, try to achieve dense-branching morphologies and dendrites. Film
the experiments and extract images of the best patterns you obtained. Print them on
A4 papers and determine their dimensions by using the Box Counting method. For this
purpose, prepare four grids, based on unit-squares having sides 2.5, 2, 1.5, 1 cm long,
respectively, and print them on transparencies.
11.14. If a spherical silver nanoparticle has a mass M and a radius r, how much do its mass increase by doubling its radius? If we have synthesized a fractal silver nanoparticle having a radius of gyration R, mass M, and dimension d = 3.5, how much do its mass increase
by doubling its radius of gyration?
11.15. The fractal dimension of a protein can be determined by calculating the mean value of the
number of alpha carbon atoms that lie within a sphere of radius R centered at an arbitrary
