Untangling Complex Systems, page 74
−
3.64886 3 6183
.
By applying the formula of the error propagation, we find that the uncertainty in δ is 0.2.
Therefore,
δ = 4 4
. ± 0 2
. in reasonable agreement with the theoretical result of 4.669…
Note that it is not an easy task that of determining the appearance of a bifurcation in the
presence of experimental noise. Usually, it is not possible to measure more than about five
period-doublings (Cvitanovic 1989). Nevertheless, the agreement between experiments
and theory is remarkable.
The Emergence of Chaos in Time
367
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0.7
0.6
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368
Untangling Complex Systems
Pattern formation
by convection
FIGURE 10.31 An example of Bénard convection with a silicone oil plus aluminum powder.
10.10. The Bénard cells are examples of what Ilya Prigogine called as dissipative structures.
A system maintained very far-from-equilibrium self-organizes, producing entropy that is
dissipated in the environment. When the Marangoni number ( Ma) is Mac < Ma ≤ 2 Mac,
the cells show a hexagonal shape (Lappa 2010). The circulation of the fluid in the hexago-
nal cells is upwards in the center and downwards along the edge. Optical investigations
have demonstrated that the surface of the fluid is depressed over the center of the cells of
a few micrometers. For Ma values larger than 2 Ma , the cells become squared. The tran-
c
sition from hexagons to squares is mediated by the occurrence of pentagonal cells. The
squared cells are more efficient in heat transfer than hexagonal cells, and their formation
is accompanied by an increase in the Nusselt number (see equation [10.23]). The patterns
that you observe also depend on the boundary conditions, i.e., the material of the plate
and its shape. An example of Bénard convection is shown in Figure 10.31 obtained with a
silicone oil having 100 mPa*s as viscosity at room temperature.
10.11. For the numerical solution of the system of nonlinear differential equations [10.36
through 10.38], you can use the ode45 solver of MATLAB. An example of script is the
following one:
function dy = lorenz(t, y)
dy = zeros(3,1);
si = 10;
r = 28;
b = 8/3;
dy(1) = si*(y(2) - y(1));
dy(2) = -y(1)*y(3) + r*y(1) - y(2);
dy(3) = y(1)*y(2) - b*y(3);
The time evolutions of variable X for the two pairs of initial conditions are shown in
Figure 10.32. The data graphed in plot (a) refer to the first two initial conditions:
[ X (0); Y (0); Z(0)] = [1 .
3
;
824
.
15
;
368
.
32 474] and [ X (0) ;′ Y (0) ;′ Z(0) ]′ = [13.
;
824 15.
;
369
.
32 474]. The data graphed in plot (b) refer to the other pair of initial conditions that are
close to S 0. It is evident that when the initial conditions are closer to S 0, the two trajectories
take more time to diverge.
By using the data of plot (a), we can calculate how the Euclidean distance δ between
the two initially very close trajectories increases over time. The output is depicted in
Figure 10.33. We confirm what we see in Figure 10.17. The ln( δ ) first grows linearly, until it saturates. The slope of the straight line fitting the growth part (see the gray straight
line in Figure 10.33) is λ ~ .
0 9.
The Emergence of Chaos in Time
369
20
10
X( t)
0
−10
(a)
0
4
8
12
16
20
24
28
t
20
10
0
X( t)
−10
−20
0
4
8
12
16
20
24
28
(b)
t
FIGURE 10.32 Trends of X( t) for two pairs of initial conditions: for the pair close to S+ in (a), and for the pair close to S 0 in (b).
4
2
0
δ||) −2
In (||
−4
−6
−8 0
5
10
15
20
t
FIGURE 10.33 Trend of ln( δ ) versus time (thin black trace). The first part of this trend has been fitted by a straight line (thick gray trace).
370
Untangling Complex Systems
In. Con.: [14.0 12.0 10.0]
In. Con.: [14.0 12.1 10.0]
10
X( t)
0
−10
0
4
8
12
16
20
10
0
Y( t) −10
−20
0
4
8
12
16
30
20
Z( t)
10
0
0
4
8
12
16
Time
FIGURE 10.34 Trends of X( t) (on top), Y( t) (in the middle), and Z( t) (at the bottom) for the two initial conditions indicated into the legend.
10.12. The solutions of the calculations obtained for the two distinct initial conditions proposed
in the text of the exercise are shown in the Figures 10.34 and 10.35.
In the first stages, both trajectories seem to describe a strange attractor. But, in the
long term, we discover that both of them spiral down toward one of the two stable solu-
tions, which are S+ and S−. What is surprising is that although the dynamic is not cha-
otic because it is not aperiodic in the long term, it shows sensitive dependence to the
initial conditions. In fact, although we start the calculations from two points that are
only δ =
0
0.1 far apart in the phase space, the system evolves towards two very distant
solutions: S+ and S−. Therefore, the behavior of the system is unpredictable, at least for
certain initial conditions, and it is referred to as transient chaos or metastable chaos or
pre-turbulence (Strogatz 1994). In the cases of transient chaos, the final states are simple:
they are periodic or stationary states. Nevertheless, the presence of a transient chaotic
stage can make the entire process hard to be predicted in its outcome. This situation is
familiar in everyday life. It suffices to think about gambling like throwing a dice or a coin.
The outcome depends sensitively on the initial orientation and velocity if the latter is suf-
ficiently large. A system exhibits transient chaos when a non-attracting chaotic saddle is
present in its phase space. The system stays near the chaotic saddle for a finite amount of
time exhibiting chaotic behavior, before exiting that region and approaching its final state
asymptotically.
10.13. In paragraph 10.7, we learned that when r > rc we can observe chaotic dynamics. In this exercise, we calculate the evolution of the Lorenz’s system for a considerable value of
The Emergence of Chaos in Time
371
35
In. Con.: [14.0 12.0 10.0]
In. Con.: [14.0 12.1 10.0]
30
25
20
Z( t) 15
10
5
Starting points
0
−15
−10
−5
0
5
10
15
X( t)
FIGURE 10.35 Trajectories traced in the Z( t)– X( t) space, starting from the two initial conditions indicated into the legend.
parameter r ( r = 350). Surprisingly, we find that the dynamic is periodic. The period of the
oscillations does not depend on the initial conditions we choose (Figures 10.36 and 10.37).
Usually, when r > rc, one finds chaotic dynamics but there are also small windows
of periodic behavior interspersed (Strogatz 1994). This phenomenology resembles that
observed in the case of the logistic map presented in paragraph 10.3.
10.14. First, I report the results achieved by following the instructions of list A. In Figure 10.38, the absorption spectra of DMA, before (trace 0), and after irradiation (traces 1 and 2) are
120
80
40
X( t)
0
−40
−80
0
2
4
6
8
10
200
0
Y( t) −200
0
2
4
6
8
10
600
400
Z( t) 200
0
0
2
4
6
8
10
Time
FIGURE 10.36 Periodic trends of X( t) (on top), Y( t) (in the middle), and Z( t) (at the bottom) obtained for σ =10, b = 8 / 3, r = 350, and [ X(0), Y(0), Z(0)] = [0, 1.00, 0] as initial condition.
372
Untangling Complex Systems
300
200
100
Y( t)
0
−100
−200
−300
−60
−30
0
30
60
90
X( t)
FIGURE 10.37 Trajectory traced in the Y( t)– X( t) space when σ = 10, b = 8 / 3, r = 350, and [ X(0), Y(0), Z(0)] = [0, 1.00, 0] is the initial condition.
2.0
1.5
IEM
2
A 1.0
400
450
500
1
0.5
λ (nm)
0
0.0 350
300
350
400
450
500
λ (nm)
FIGURE 10.38 Absorption spectra of DMA before (continuous black trace, labeled as 0) and after two dis-
tinct irradiations (dotted light gray and dashed gray traces, labeled as 1 and 2, respectively). The fluorescence
spectrum of DMA is shown in the inset.
reported. The spectra 1 and 2 have been recorded after two distinct irradiation experi-
ments. They are significantly different from spectrum 0. This evidence suggests that UV
irradiation at 260 nm triggers reactions that decompose DMA. In fact, it is known that
when anthracene derivatives, substituted in positions 9 and/or 10, are excited in their
higher energy optically active transition (i.e., below 300 nm), they participate in photo-
dissociation reactions. In other words, they release their substituents as radical groups
(Favaro et al. 2007). Moreover, the radiation at 260 nm is absorbed not only by DMA
but also the solvent CHCl . Upon irradiation at 260 nm, chloroform produces chlorine
3
radicals. Then, chlorine radicals induce chain reactions involving oxygen and promoting
a more or less noticeable oxidative degradation of DMA (Laplante and Pottier 1982).
The Emergence of Chaos in Time
373
(1) λexc = 260 nm, λem = 408 nm, h = 3.5 cm
(2) λexc = 260 nm, λem = 408 nm, h = 3.5 cm
(3) λexc = 260 nm, λem = 408 nm, h = 1.2 cm
1
.u.)
.u.)
(a
2
(a
I em
I em
1
3
0
500 1000 1500 2000 2500 3000 3500
850 900 950 1000 1050 1100 1150
(a)
t (sec)
(b)
t (sec)
2
3
.u.)
.u.)
(a
(a
I em
I em
400 450 500 550 600 650 700 750 800 850 900
260
280
300
320
340
(c)
t (sec)
(d)
t (sec)
FIGURE 10.39 Time evolutions of the DMA emission (recorded at λ
em = 408 nm) upon steady irradia-
tion at 260 nm recorded in three experiments labeled as 1, 2 and 3, respectively. The conditions of the three
experiments are indicated in the legend of graph (a), wherein an overall image of the three kinetics is shown.
Enlargements of 1 in (b), 2 in (c) and 3 in (d).
Examples of DMA fluorescence signals that can be recorded upon steady irradiation at
260 nm are shown in Figure 10.39.
Graph (a) of Figure 10.39 shows the time evolutions of DMA fluorescence collected at
408 nm in three distinct experiments, numbered consecutively 1, 2 and 3. Experiments
1 and 2 are two replicas. In fact, the UV irradiation was carried out only at the bottom
of 3.5 mL. On the other hand, in case 3, all the 1.2 mL of the solution were under UV.
In all the three experiments, the luminescence intensity decreased because DMA was
degraded. If we zoom in on every kinetics (see the other three graphs of the same figure),
we appreciate their differences. Only in traces 1 and 2, we see luminescence oscillations.
In 3, they are absent, because there was no convection. The oscillations are just due to the
convective motions of the solution, and they do not depend on the photo-induced reac-
tions. In fact, the oscillations recorded in 1 and 2 can also be observed in the light scat-
tered by the solution, and they disappear under stirring (Laplante et al. 1984). Although
experiments 1 and 2 were performed in the same conditions, they originated two signifi-
cantly different results. This behavior is the “butterfly effect” in action. We quantify the
difference between the two traces by calculating their Fourier spectra (see Figure 10.40).
The Fourier spectrum of trace 1 consists of just one fundamental frequency that is
ν0 = 0.0586 s−1 and corresponds to a period of 17 s. The other peaks at 0.1172 s−1 (i.e.,
2ν0 ) and 0.1758 s−1 (i.e., 3ν0 ) are the second and third harmonics, respectively. The
Fourier spectrum of trace 2 is more complicated than that of trace 1. It has the largest
374
Untangling Complex Systems
1
Amplitude
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Frequency (Hz)
2
Amplitude
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Frequency (Hz)
FIGURE 10.40 Fourier spectra of the fluorescence kinetics 1 (graph on top) and 2 (graph at the bottom)
shown in Figure 10.39.
amplitude at the frequency of 0.0078 s−1 that corresponds to a period of 128 s. There
is also another important frequency of 0.043 s−1 that corresponds to a period of 23 s.
Based on the kinetic traces and their Fourier spectra, we infer that the Rayleigh numbers
of experiments 1 and 2 were both included between Ra and Ra , but Ra for 2 was
c,1
c,2
slightly larger than Ra for 1. Therefore, from the definition of Ra (equation [10.48]), it
