Untangling Complex Systems, page 72
the book edited by Kantz et al. (1998), which focused on physiological data, and that by
Abarbanel (1996). Another text that presents the role of chaos in physiological rhythms is
that by Glass and Mackey (1988). A book dealing with the role of Chaos in the brain is that
edited by Lehnertz et al. (2000).
• Schöll and Schuster (2008) have edited a handbook for whoever is interested in chaos control.
• The idea that nonlinear dynamics can be “message sources” has been developed also by
Crutchfield and Young (1989), and Crutchfield (1994).
10.13 EXERCISES
10.1. Include all the following equations in two sets: the set of linear equations and the
set of nonlinear equations. ( a) sin x = y; ( b) ( dy / dt) + y + lny = 0; ( c) (
2 dy/ dt) + 3 y = ;
0
( d) (
5 d 2 y/ dt 2) − ( dy/ dt) + 2 y = ;
0 ( e) (
5 d 2 y/ dt 2)2 − ( dy/ dt) + 2 y = ;
0 ( f ) x 2 + x + 2 = y;
( g) 2 x + 3 = y; ( )
h ( dy dt) − xy + 2 x = ;
0 ( i) y = lnx − x.
10.2. Via numerical integration (using, e.g., the MATLAB ode45 solver), determine the evolution
of the double pendulum for θ (
(
θ1( t 0) = ω1 = ,
1 t = 0) = −0.100, θ 2 t = 0) = 0.520, and ′
=
0
θ′2 ( t = 0) = ω2 = 0. The motion (Shinbrot et al. 1992) is expressed by the following first order
equations:
θ′1 = ω1
θ′2 = ω2
g ( s θ
2
2
in 2 cos( θ
∆ ) − µ si θ
n 1) − ( L θ
2 ′2 + L θ
1 ′
1 cos ( θ
∆ ))sin( θ
∆ )
ω′1 =
(
cos 2 ∆ ))
1
L µ −
( θ
µ ( θ
2
2
g
sin 1 cos( θ
∆ ) − si θ
n 2 ) + (µ Lθ1′1 + L θ
2 ′2 cos ( θ
∆ ))sin( θ
∆ )
ω′2 =
L
2
( cos ∆ ))
2 µ −
( θ
where
∆θ = θ1 −θ2 and µ = ( m 2 + m 1)/ m 2.
Compare the solution with those shown in Figure 10.3.
10.3. Via numerical integration (using, e.g., the MATLAB ode45 solver), determine the evolu-
tion of the double pendulum for θ (
(
1 t = 0) = 1.57, θ 2 t = 0) = 1.57 (case (a) in Figure 10.22),
and θ′
θ2 ( t 0)
1 ( t = 0) = 0, ′
=
= 0. For the equations of motion see exercise 10.2. Compare the
solution of this exercise with the behavior obtained in the 3° run of Figure 10.4 (its initial
conditions are shown in graph (b) of Figure 10.22).
10.4. Integrate the logistic equation [10.10].
10.5. Use the logistic map and determine the evolution of two populations that start from two
slightly different initial conditions: x′0 = 0 30000
.
and x′′0 = 0 30001
.
, respectively. Use two
different values of growth rate: (a) r = 3, and (b) r = 4. What differences do you ascertain
in the two cases? Calculate the abundance of the populations for, at least, 50 iterations. You
can use either a spreadsheet or MATLAB or any other software for the calculations.
The Emergence of Chaos in Time
357
y
y
m 1
m 2
m 2
m 1
x
x
(a)
(b)
FIGURE 10.22 (a and b) Two different initial conditions for the double pendulum.
10.6. Another example of a unimodal map is x
π
≤ r ≤
≤ x ≤
1
( )
n+ = rsin
xn , with 0
1 and 0
1.
Verify that it is a unimodal map and generate the bifurcation diagram (e.g., by using
MATLAB). Compare this latter bifurcation diagram with that built for the logistic map.
10.7. Apply equation [10.17], defining the Feigenbaum’s constant, to estimate the value of r
at which the logistic map becomes chaotic ( r∞). For the calculation, use the coordinates
of the following two bifurcation points: a cycle with period 16 is at rk−1 = 3.564407, and
the subsequent bifurcation point, originating a cycle with period 32, is at r k= 3.568750.
Calculate r
k+ 1 ≈ r∞.
10.8. Libchaber investigated the transition of the convective motion from the laminar to the tur-
bulent regime by using liquid helium at 3 K. By knowing that the isobaric thermal expan-
sion coefficient is β = 6.15 × 10−2 K−1 (Lide 2005), dynamic viscosity μ = 2.7 × 10−4 cm2 s−1,
thermal diffusivity α = 4.3 × 10−4 cm2 s−1 (Libchaber and Maurer 1978), g = 9.8 m2 s−1, and
reminding that the critical Ra value for observing spontaneous formation of cylindrical
convective rolls is 1708, calculate the thermal gradient needed when the height of the cell
is 0.1 cm and when it is 1 cm long.
10.9. In the experiments described by Libchaber et al. (1982), the authors increased constantly
the thermal gradient between the two plates of the cell. They wanted to confirm the phe-
nomenon of the period doubling cascade as a route to chaos as predicted for the unimodal
maps by Feigenbaum. Using equation [10.17], and knowing that the critical values of the
ratio Ra / Ra for k
k
c
= 2, 3 and 4 were 3.4850, 3.6183 and 3.6486, respectively, calculate the
value of the parameter δ. Estimate the final uncertainty by using the propagation formula
for the “maximum a priori absolute error” (see Appendix D) knowing that the uncertainty
in each ratio ( Ra / Ra ) is ±0 0005
.
.
k
c
10.10. This exercise consists in observing “in vivo” the Bénard cells described in Box 10.1 of
this chapter. Go to your wet laboratory, wear a white coat, gloves, safety glasses, and
recover the following equipment: a hot plate, a small metallic pan, aluminum powder
(or bronze powder) and silicone oil. There are many types of silicone oils, which are all
polysiloxanes, thus, polymers having a chain with alternating silicon and oxygen atoms,
and with organic side chains. They cover a broad range of viscosities. It is convenient to
use a silicone oil with a dynamic viscosity of at least 0.5 cm2/s (Van Hook and Schatz
1997). Mix the aluminum powder and the oil. Pour the mixture into the metallic pan to
form a thin layer of fluid (around 1 mm thick). Heat the pan gently, and observe the pattern
formation (it could be enough to heat the layer between 50°C and 100°C). You do not need
very high temperatures to see Bénard cells. It is important to avoid burning your oil. After
observing a pattern, destroy it by shaking the mixture. Then leave the system alone and
look at the recovery of the pattern. What does it mean? What is the shape of the Bénard
cells? Does the shape of the cells depend on the thermal gradient?
10.11. Solve the system of three differential equations [10.36–10.38] of Lorenz’s model numerically
after fixing σ = 10, b = 8 3
/ , r = 28. Start the calculations by using two slightly different
358
Untangling Complex Systems
initial conditions that are both close to the steady state solution S+: [ X (0); Y (0); Z(0)] =
[13.824; 15.368; 32.474] and [ X (0)′; Y (0)′; Z(0)′] = [13.
;
824 15.
;
369 32.474]. Perform
the calculation in the time interval [0–30]. Calculate (I) how the Euclidean distance δ ( t)
between the two trajectories changes over time, and (II) the Lyapunov exponent. Repeat the
calculations using another pair of initial conditions that are close to the steady solution S 0:
[ X (0)′′; Y (0)′′; Z(0)′′] = [ ;
0 .
1
;
000 0] and [ X (0) ;
′′′ Y (0) ;
′′′ Z(0) ]
′′′ = [ ;
0 1.
;
001 0]. What is the
difference between the two situations?
10.12. Solve the system of nonlinear differential equations [10.36–10.38] of the Lorenz’s model
numerically after fixing σ = 10, b = 8 3
/ and r = 20. Start the calculations by using two
initial conditions that differ slightly: [14.0 12.0 10.0] and [14.0 12.1 10.0]. What do you
observe?
10.13. Solve the system of nonlinear differential equations [10.36–10.38] of the Lorenz’s model
numerically after fixing σ = 10, b = 8 3
/ and r = 350. Start the calculations by using ini-
tial conditions that differ appreciably. What do you observe?
10.14. In this exercise, you make an experience of the “butterfly effect” in action. Go to your
wet laboratory, wear a white coat, gloves and safety glasses. If you have a spectrofluo-
rimeter in your laboratory, you can use 9,10-dimethylanthracene (DMA) as a probe of
the convective motions and follow the list A of instructions. On the other, if you have a
UV-visible spectrophotometer, but also a UV lamp for irradiation, you can use a photo-
chromic compound and follow the list B of instructions. The photochromic species could
be the spirooxazine depicted in Figure 10.18 or any other compound as long as it colors
at room temperature under the photon flux of your lamp. The two possible experimental
set-ups are sketched in Figure 10.23, wherein I is the light emitted by DMA, whereas
EM
I
is the intensity of the probe ray emitted by the lamp of the spectrophotometer in case
probe
we use a photochromic compound.
List A of instruction.
A –Prepare at least 10 mL of a 9,10-dimethylanthracene (DMA) solution in chloroform
1 having a concentration of about 4 × 10−5 M.
A –Pour 3 or slightly more milliliters of the solution into a fluorometric cuvette having
2 1 cm as path length. Measure the height of the solution (it should be ≥3 cm).
A –Record the UV-visible absorption spectrum of the solution.
3
A –Collect its emission spectrum by exciting at 260 nm (select the widths of the slits and
4 the time integration to avoid saturation of the detector).
A –Maintaining the cuvette inside the sample holder of the fluorimeter, uncap it, and
5 wait for a time interval of 10′–15′. Use a timer to determine the exact period waited
UV
UV
I EM
I probe
(a)
(b)
FIGURE 10.23 Experimental set-up when we use a fluorescent probe in (a) or a photochromic probe in (b).
The Emergence of Chaos in Time
359
before executing the next step. Meanwhile, monitor the environmental conditions in
your room, by measuring temperature, pressure, and humidity.
A –Run an experiment wherein you collect how the intensity emitted by DMA, at a spe-
6 cific wavelength, changes over time (for at least 2000 s) under continuous irradiation at
260 nm. Store data every 1 s after fixing the integration time to 1 s.
A –At the end of the experiment, cap the cuvette and measure the final height of the solu-
7 tion. Did the solution evaporate? How much? Estimate the power ( Pw) absorbed by the
solution through the following formula: Pw = H
∆
⋅ ρ ∆ (
)
vap
⋅ V MW ⋅ t
∆ , where Δ H is
vap
the heat of vaporization (31.4 kJ/mol), ρ the density (1.489 g/cm3), and MW the molecu-
lar weight (119.38 g/mol) of chloroform. Δ V is the amount of volume that evaporated,
and Δ t is the time when the cuvette was maintained uncapped.
A –Collect the absorption spectrum of the solution. What do you observe? Did it change
8 respect to the spectrum recorded at point A ?3
A –Repeat the instructions from A up to A by using a fresh solution of DMA in chlo-
9
2
8
roform. Do you observe perfect reproducibility, especially for the fluorescence signal
recorded versus time?
A –Based on the profile of the fluorescence signal versus time, and the number of frequen-
10cies determined by calculating its Fourier Transform (see Appendix C), try to estimate
the vertical thermal gradient by using the formula of the Rayleigh number [10.48]. For
chloroform, β = 1.107 × 10−3 K−1, μ = 4 × 10−3 cm2 s−1, α = 7.31 × 10−4 cm2 s−1.
List B of instructions.
B –Prepare at least 10 mL of the photochromic solution in acetone as the solvent, and at a
1 concentration of about 9 × 10−5 M.
B –Pour 3 or slightly more milliliters of the solution into a fluorometric cuvette having
2 1 cm as path length. Measure the height of the solution (it should be ≥3 cm).
B –Maintaining the cuvette inside the sample holder of the UV-visible spectrophotom-
3 eter, uncap it and wait for a time interval of 10′–15′. Use a timer to determine the exact
time waited before executing the next step. Meanwhile, turn on the lamp for irradiation
and monitor the environmental conditions in your room by measuring temperature,
pressure, and humidity.
B –The UV irradiation must be performed at 90
4
° respect to the spectrophotometric rays.
You can convey the UV radiation to the bottom part of the fluorometric cell through an
optical fiber, as it is shown in Figure 10.19. If your spectrophotometer has a diode array
as the detector, you can record the complete UV-visible spectra during irradiation at
every single shot. Otherwise, if you have a traditional double- or single-beam spectro-
photometer, with a photomultiplier as the detector, you should know the wavelength
that corresponds to the maximum for the band into the visible, and then tune your
monochromator to that wavelength. Run an experiment and record how the UV-visible
spectrum or the absorbance at the selected wavelength evolves under constant irradia-
tion in the UV. Store data every 3 s, and for at least one hour.
B –At the end of the experiment, cap the cuvette and measure the final height of the solu-
5 tion. Did the solution evaporate? How much? Estimate the power ( Pw) absorbed by the
solution through the following formula: Pw = H
∆
⋅ ρ ∆ (
)
vap
⋅ V MW ⋅ t
∆ , where Δ H is
vap
the heat of vaporization (32.1 kJ/mol), ρ the density (0.7845 g/cm3), and MW the molec-
ular weight (58.08 g/mol) of acetone. Δ V is the amount of volume that evaporated, and
Δ t is the time interval wherein the cuvette was maintained uncapped.
B –Repeat the instructions from B up to B by using a fresh solution of the photochromic
6
2
5
compound in acetone. Do you observe perfect reproducibility, especially for the signal
of absorbance vs. time?
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Untangling Complex Systems
B –Based on the profile of the absorbance vs. time and the number of frequencies deter-
7 mined by calculating its Fourier spectrum (see Appendix C), try to estimate the verti-
cal thermal gradient by using the formula of the Rayleigh number [10.48]. For acetone,
β = 1.43 × 10−3 K−1, μ = 4 × 10−3 cm2 s−1, α = 9.48 × 10−4 cm2 s−1.
10.15. The nonlinear time series analysis is being successfully applied in a variety of disciplines.
For instance, in medicine. In fact, clinical diagnoses depend on the ability to record and
analyze physiological signals, such as electrocardiograms (ECG), electroencephalo-
grams (EEG), heart rate recordings, concentrations of hormones, et cetera. Typically,
these signals are generated by processes that are nonlinear and nonstationary. An analy-
sis of their features promises to be of clinical value, distinguishing states of normal or
pathological functioning, and forecasting worsening conditions. The National Center for
Research Resources of the National Institutes of Health has created a Research Resource
for Complex Physiologic Signals (Goldberger et al. 2000). The resource consists of three
elements. One is PhysioBank that is an archive of physiological signals. The second is
PhysioToolkit that is a library of open-source software for physiological signal process-
ing and analysis. The third is PhysioNet that is an online forum for the dissemination and
exchange of biomedical signals and open-source software. In PhysioBank, you can find
many time series, and you can play by analyzing their nonlinear features. For example,
you can download a dataset of heart rate and chest volume recorded from a patient in the
sleep laboratory of the Beth Israel Hospital in Boston (MA, USA) (Rigney et al. 1993;
