Introduction to CHAOS

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Introduction to CHAOS

• Larry Liebovitch, Ph.D.
• Florida Atlantic University
• 2004

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These two sets of data have the same

• mean
• variance
• power spectrum

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RANDOM random x(n) RND
Data 1
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CHAOS Deterministic x(n1) 3.95 x(n) 1-x(n)
Data 2
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etc.
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RANDOM random x(n) RND
Data 1
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CHAOS deterministic x(n1) 3.95 x(n) 1-x(n)
Data 2
x(n1)
x(n)
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CHAOS
Definition
predict that value
Deterministic
these values
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CHAOS
Definition
Small Number of Variables
x(n1) f(x(n), x(n-1), x(n-2))
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CHAOS
Definition
Complex Output
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CHAOS
Properties
Phase Space is Low Dimensional
d , random
d 1, chaos
phase space
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CHAOS
Properties
Sensitivity to Initial Conditions
nearly identical initial values
very different final values
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CHAOS
Properties
Bifurcations
small change in a parameter
one pattern
another pattern
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Time Series
X(t)
Y(t)
Z(t)
embedding
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Phase Space
Z(t)
phase space set
Y(t)
X(t)
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Attractors in Phase Space
Logistic Equation
X(n1) 3.95 X(n) 1-X(n)
X(n1)
X(n)
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Attractors in Phase Space
Z(t)
Lorenz Equations
Y(t)
X(t)
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The number of independent variables smallest
integer gt the fractal dimension of the attractor
Logistic Equation
phase space
dlt1
time series
X(n1)
X(n)
d lt 1, therefore, the equation of the time series
that produced this attractor depends on 1
independent variable.
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The number of independent variables smallest
integer gt the fractal dimension of the attractor
Lorenz Equations
phase space
time series
d 2.03
Z(t)
X(n1)
n
X(t)
Y(t)
d 2.03, therefore, the equation of the time
series that produced this attractor depends on 3
independent variables.
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Data 1
time series
phase space
d
Since , the time series was
produced by a random mechanism.
d
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Data 2
time series
phase space
d 1
Since d 1, the time series was produced by a
deterministic mechanism.
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Phase Space
Constructed by direct measurement
Measure X(t), Y(t), Z(t)
Z(t)
Each point in the phase space set has
coordinates X(t), Y(t), Z(t)
X(t)
Y(t)
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Phase Space
Constructed from one variable
Takens Theorem Takens 1981 In Dynamical Systems
and Turbulence Ed. Rand Young,
Springer-Verlag, pp. 366 - 381
X(t)
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Position and Velocity of the Surface of a Hair
Cell in the Inner Ear
Teich et al. 1989 Acta Otolaryngol (Stockh),
Suppl. 467 265 - 279
10-1
stimulus 171 Hz
velocity (cm/sec)
-10-1
3 x 10-5
displacement (cm)
-10-4
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Position and Velocity of the Surface of a Hair
Cell in the Inner Ear
Teich et al. 1989 Acta Otolaryngol (Stockh),
Suppl. 467 265 - 279
3 x 10-2
stimulus 610 Hz
velocity (cm/sec)
-3 x 10-2
displacement (cm)
5 x 10-6
-2 x 10-5
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RANDOM x(n) RND
Data 1
fractal demension of the phase space set
fractal dimension of phase space set
embedding dimension number of values of the
data taken at a time to produce the phase space
set
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Data 2
CHAOS deterministic x(n1) 3.95 x(n) 1 - x(n)
fractal demension of the phase space set 1
fractal dimension of phase space set
embedding dimension number of values of the
data taken at a time to produce the phase space
set
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Chick Heart Cells
microelectrode
Glass, Guevara, Bélair Shrier. 1984 Phys. Rev.
A291348 - 1357
v
voltmeter
current source
chick heart cell
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Chick Heart Cells
Spontaneous Beating, No External Stlimulation
voltage
time
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Chick Heart Cells
Periodically Stimulated 2 stimulations - 1 beat
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Chick Heart Cells
Periodically Stimulated 1 stimulation - 1 beat
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Chick Heart Cells
Periodically Stimulated 2 stimulations - 3 beats
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The Pattern of Beating of Chick Heart Cells
Glass, Guevara, Bélair Shrier.1984 Phys. Rev.
A291348 - 1357
periodic stimulation - chaotic response
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The Pattern of Beating of Chick Heart Cells
continued
phase of the beat with respect to the stimulus
phase vs. previous phase
experiment
theory (circle map)
1.0
0.5
i 1
0
0.5
1.0
0
1.0
0.5
i
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The Pattern of Beating of Chick Heart Cells
Glass, Guevara, Belair Shrier.1984 Phys. Rev.
A291348 - 1357
Since the phase space set is 1-dimensional, the
timing between the beats of these cells can be
described by a deterministic relationship.
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Procedure

• Time series
• e.g. voltage as a function of time
• Turn the Time Series into a Geometric Object
• This is called embedding.

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Procedure

• Determine the Topological Properties of this
  Object
• Especially, the fractal dimension.
• High Fractal Dimension
• Random chance
• Low Fractal Dimension
• Chaos deterministic

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The Fractal Dimension is NOT equal to The
Fractal Dimension
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Fractal DimensionHow many new pieces of the
Time Series are found when viewed at finer time
resolution.
d
X
time
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Fractal DimensionThe Dimension of the Attractor
in Phase Space is related to theNumber of
Independent Variables.
x(t2 t)
d
X
x(t)
x(t t)
time
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Mechanism that Generated the Data
Chance d(phase space set)
Data
?
x(t)
Determinism d(phase space set) low
t
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Lorenz1963 J. Atmos. Sci. 2013-141
(Rayleigh, Saltzman)
Model
C O L D
HOT
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Lorenz1963 J. Atmos. Sci. 2013-141
Equations
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Lorenz1963 J. Atmos. Sci. 2013-141
Equations

• X speed of the convective circulation
  X gt 0 clockwise,
  X lt 0 counterclockwise
• Y temperature difference between rising and
  falling fluid

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Lorenz1963 J. Atmos. Sci. 2013-141
Equations

• Z bottom to top temperature minus the linear
  gradient

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Lorenz1963 J. Atmos. Sci. 2013-141
Phase Space
Z
X
Y
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Lorenz Attractor
cylinder of air rotating counter-clockwise
cylinder of air rotating clockwise
X lt 0
X gt 0
50
Sensitivity to Initial ConditionsLorenz Equations
Initial Condition
X 1.
X(t)
0
different
same
X 1.00001
X(t)
0
IXtop(t) - Xbottom(t)I e t Liapunov
Exponent
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Deterministic, Non-Chaotic
X(n1) f X(n)
Accuracy of values computed for X(n)
1.736 2.345 3.254 5.455 4.876
4.234 3.212
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Deterministic, Chaotic
X(n1) f X(n)
Accuracy of values computed for X(n)
3.455 3.45? 3.4?? 3.??? ? ? ?
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Initial Conditions X(t0), Y(t0), Z(t0)...
Clockwork Universe determimistic non-chaotic
Can compute all future X(t), Y(t), Z(t)...
Equations
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Initial Conditions X(t0), Y(t0), Z(t0)...
Chaotic Universe determimistic chaotic
sensitivity to initial conditions
Can not compute all future X(t), Y(t), Z(t)...
Equations
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Lorenz Strange Attractor
starting away
Trajectories from outside pulled TOWARDS it why
its called an attractor
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Lorenz Strange Attractor
starting on
Trajectories on the attractor pushed APART from
each other sensitivity to initial conditions
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Strangeattractor is fractal
phase space set
not strange
strange
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Chaoticsensitivity to initial conditions
time series
X(t)
X(t)
t
t
not chaotic
chaotic
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Shadowing Theorem
If the errors at each integration step are small,
there is an EXACT trajectory which lies within a
small distance of the errorfull trajectory that
we calculated
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Shadowing Theorem
There is an INFINITE number of trajectories on
the attractor. When we go off the attractor, we
are sucked back down exponentially fast. Were on
an exact trajectory, just not on the one we
thought we were on.
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4. We are on a real trajectory.
3. Pulled back towards the attractor.
2. Error pushes us off the attractor.
1. We start here.
Trajectory that we are trying to compute.
Trajectory that we actually compute.
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Sensitivity to initial conditions means that the
conditions of an experiment can be quite similar,
but that the results can be quite different.
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TUESDAY
10 µl
ArT
64

WEDNESDAY
10 µl
ArT
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X(n 1) A X(n) 1 -X (n)
A 3.22
X(n)
n
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X(n 1) A X(n) 1 -X (n)
A 3.42
X(n)
n
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Bifurcation
A 3.62
X(n)
n
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x(n 1) A x(n) 1 -x(n)

• Start with one value of A.
• Start with x(1) 0.5.
• Use the equation to
• compute x(2) from x(1).
• Use the equation to
• compute x(3) from x(2) and
• so on... up to x(300).

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x(n 1) A x(n) 1 -x(n)

• Ignore x(1) to x(50), these
• are the transient values off
• of the attractor.
• Plot x(51) to x(300) on the
• Y-axis over the value of A
• on the X-axis.
• Change the value of A, and
• repeat the procedure again.

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Sudden changes of the pattern indicate
bifurcations ( )
x(n)
x(n)
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Glycolysis
The energy in glucose is transfered to ATP. ATP
is used as an energy source to drive biochemical
reactions.
-

-
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Glycolysis
Theory Markus and Hess 1985 Arch. Biol. Med. Exp.
18261-271
sugar input
ATP output
periodic
time
time
chaotic
time
time
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Glycolysis
Experiments Hess and Markus 1987 Trends. Biomed.
Sci. 1245-48
cell-free extracts from bakers yeast
ATP measured by fluorescence glucose input
time
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Glycolysis
Periodic
fluorescence
Experiments Hess and Markus 1987 Trends. Biomed.
Sci. 1245-48
Vin
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Glycolysis
Experiments Hess and Markus 1987 Trends. Biomed.
Sci. 1245-48
Chaotic
20 min
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Glycolysis Markus et al. 1985. Biophys. Chem
2295-105
Bifurcation Diagram
theory
experiment
chaos
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Glycolysis Markus et al. 1985. Biophys. Chem
2295-105
ADP measured at the same phase each time of the
input sugar flow cycle (ATP is related to ADP)
period of the ATP concentration

period of the input sugar flow cycle
frequency of the input sugar flow cycle
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Phase Transitions Haken 1983 Synergetics An
Introduction Springer-Verlag Kelso 1995
Dynamic Patterns MIT Press
Try to tap the right index finger out-of-phase
with the tick of the metronome.
Tap the left index finger in-phase with the
tick of the metronome.
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Phase Transitions Haken 1983 Synergetics An
Introduction Springer-Verlag Kelso 1995
Dynamic Patterns MIT Press
As the frequency of the metronome increases, the
right finger shifts from out-of-phase to in-phase
motion.
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Phase Transitions Haken 1983 Synergetics An
Introduction Springer-Verlag Kelso 1995
Dynamic Patterns MIT Press
A. TIME SERIES
ABD
ADD
Position of Right Index Finger Position of Left
Index Finger
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Self-Organized Phase Transitions Haken 1983
Synergetics An Introduction Springer-Verlag
Kelso 1995 Dynamic Patterns MIT Press
B. POINT ESTIMATE OF RELATIVE PHASE
360o
180o
2 sec
0o
Position of Right Index Finger
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Phase Transition Haken 1983 Synergetics An
Introduction Springer-Verlag Kelso 1995 Dynamic
Patterns MIT Press
This bifurcation can be explained as a change in
a potential energy function similar to the change
which occurs in a physical phase transition.
system potential
scaling parameter
83

Small changes in parameters can produce large
changes in behavior.
10cc ArT

9cc ArT
84
Bifurcations can be used to test if a system is
deterministic.
Deterministic Mathematical Model
Experiment
predicted bifurcations
observed bifurcations
Match ?
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The fractal dimension of the phase space set
tells us if the data was generated by a random or
deterministic mechanism.
Experimental Data
x(t)
t
86
The fractal dimension of the phase space set
tells us if the data was generated by a random or
a deterministic mechanism.
Phase Space Set
X(t)
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The fractal dimension of the phase space set
tells us if the data was generated by a random or
a deterministic mechanism.
d low
d
Deterministic
Random
Mechanism that generated the experimental data.
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Epidemics Schaffer and Kot 1986 Chaos ed. Holden,
Princeton Univ. Press
New York
chickenpox
measles
4000
15000
time series
0
0
phase space
89
Epidemics Olsen and Schaffer 1990 Science
249499-504 dimension of attractor in phase space
chickenpox
measles
Kobenhavn 3.1 3.4 Milwaukee
2.6 3.2 St. Louis
2.2 2.7 New York 2.7
3.3
90
Epidemics Olsen and Schaffer 1990 Science
249499-504
SEIR models - 4 independent
variables S susceptible
E exposed, but not yet
infectious
I infectious R
recovered
91
Epidemics Olsen and Schaffer 1990 Science
249499-504
Conclusion measles chaotic
chickenpox noisy yearly cycle
92
Electrocardiogram ECG Electrical recording of
the muscle activity of the heart.
time series voltage Kaplan and Cohen 1990 Circ.
Res. 67886-892
fibrillation death
normal
Phase space V(t), V(t t)
8
D random
D 1 chaos
93
Electrocardiogram ECG Electrical recording of
the muscle activity of the heart.
time series voltage Babloyantz and Destexhe 1988
Biol. Cybern. 58203-211
normal
D 6 chaos
94
Electrocardiogram ECG Electrical recording of
the muscle activity of the heart.
normal
time series time between heartbeats Babloyantz
and Destexhe 1988 Biol. Cybern. 58203-211
D 6 chaos
fibrillation death
Evans, Khan, Garfinkel, Kass, Albano, and Diamond
1989 Circ. Suppl. 80II-134
D 4 chaos
induced arrhythmias
Zbilut, Mayer-Kress, Sobotka, OToole and Thomas
1989 Biol. Cybern, 61371-381
D 3 chaos
95
Electroencephalogram EEG Electrical recording of
the nerve activity of the brain. Mayer-Kress and
Layne 1987 Ann. N.Y. Acad. Sci. 50462-78
time series V(t)
phase space
V(t)
V(t t)
D8 chaos
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Electroencephalogram EEG Electrical recording of
the nerve activity of the brain.
Rapp, Bashore, Martinerie, Albano, Zimmerman, and
Mees 1989 Brain Topography 299-118
Babloyantz and Destexhe 1988 In From Chemical to
Biological Organization ed. Markus, Muller, and
Nicolis, Springer-Verlag
Xu and Xu 1988 Bull. Math. Biol. 5559-565
97
Electroencephalogram EEG Electrical recording of
the nerve activity of the brain.
Different groups find different dimensions under
the same experimental conditions.
98
Electroencephalogram EEG Electrical recording of
the nerve activity of the brain.
perhaps
mental task quiet awake, eyes closed quiet
sleep brain virus Creutzfeld- Jakob Epilepsy
petit mal meditation Qi-kong
High Dimension
Low Dimension
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Random Markov
How to compute the next x(n) Each t pick a
random number 0 lt R lt 1 If open, and R lt pc,
then close. If closed, and R lt po, then
open.
100
Random Markov
If open probability to close in the next t pc
open
If closed probability to open in the next tpo
t
closed
101
Deterministic Iterated Map Liebovitch Tóth 1991
J. Theor. Biol. 148243-267
open
x(n1)
closed
x(n)
x(n) the current at time n x(n1) f (x(n))
102
Deterministic Iterated Map Liebovitch Tóth 1991
J. Theor. Biol. 148243-267
How to compute the next x(n)
x(3)
x(2)
0
0
x(2)
0
x(1)
0
103
Tacoma Narrows Bridge
Thursday November 7, 1940
Good modern review (explaining why the
explanation given in physics textbooks is
wrong) Billah and Scanlan 1991 Am. J. Phys.
59118-124
104
Tacoma Narrows Bridge
Equation of flutter that destroyed the Tacoma
Narrows Bridge x Ax Bx f ( x, x )
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Tacoma Narrows Bridge
Wind Tunnel Tests
Scanlan and Vellozzi 1980 in Long Span Bridges
ed. Cohen and Birdsall pp. 247-263 NYAS
ORIGIONAL TACOMA NARROWS
AIRFOIL
(
106
Tacoma Narrows Bridge
Wind Tunnel Tests
The drag on an airplane wing (A) increases with
wind speed.
0.3
A2

The drag on the OTN (original Tacoma Narrows)
bridge changes sign as the wind speed increases,
it enters into positive feedback.
OTN
0.2
0.1
U NB
0
0.1
A
0.2
107
Random
Like a small molecule, relentlessly kicked by
the surrounding heat from one state to another.
The change of states is driven by chance kT
thermal fluctuations.
OPEN
CLOSED
random
energy
108
Deterministic
Like a lilttle mechanical machine with sticks
and springs.
The change of states is driven by coherent
motions that result from the structure and the
atomic, electrostatic, and hydrophobic forces
in the channel protein.
OPEN
CLOSED
energy
deterministic
109
Analyzing Experimental Data
The Good News
In principle, you can tell if the data was
generated by a random or a deterministic
mechanism.
110
Analyzing Experimental Data
The Bad News
In practice, it isnt easy.
111
Why its Hard to Tell Random from Deterministic
Mechanisms
Need Lots of Data

• Very large data sets 10d?
• Sampling rate must cover the
• attractor evenly.
• Sample too often only see 1-d trajectories.
• Sample too rarely dont see the attractor at
  all.

112
Why its Hard to Tell Random from Deterministic
Mechanisms
Analyzing the Data is Tricky

• Choice of lag time t for the embedding.
• lag too small the variable doesnt change
  enough, derivatives not accurate.
• lag too long the variable changes too
  much, derivatives not accurate.
• Method of evaluating the dimension.

113
Why its Hard to Tell Random from Deterministic
Mechanisms
Mathematics is Not Known

• Embedding theorems are only proved for smooth
  time series.

114
How Many Times Series Values?
N Number of values in the time series needed
to correctly evaluate the dimension of an
attractor of dimension D
N when D 6
115
How Many Times Series Values?
Smith 1988 Phys. Lett. A133283 42D
5,000,000,000
Wolff et al. 1985 Physica D16285 30D
700,000,000
Wolf et al. 1985 Physica D16285 10D
1,000,000
116
How Many Times Series Values?
Nerenberg Essex 1990
Phys. Rev. A427065
_______1________ kd1/2A In (k)(D2)/2
D2 2
200,000
D/2
2(k-1) ((D4)/2) (1/2) ((D3)/2)
x

117
How Many Times Series Values?
Ding et al. 1993 Phys. Rev. Lett. 703872
10D/2
1,000
(D/2)! D/2
Gershenfeld 1990 preprint
2D
10
118
Lorenz
X(t)
119
Lorenz
t correlation time
X(t)
120
Lorenz
X(t)
121
Takens Theorem
If
122
Then, the lag plot constructed from the data
X(t t)
X(t)
123
Is a linear transformation of the real phase
space
dX(t) dt
X(t)
124
because
dX(t) dt
X(t t) - X(t) t
125
Since the fractal dimension is invariant under a
linear transformation, the fractal dimension of
the lag plot is equal to the fractal dimension of
the real phase space set.
126
Takens Theorem
If the data does not satisfy these assumptions
then we are not guaranteed that the fractal
dimension of the lag plot is equal to the fracfal
dimension of the real phase space set.
127
For example
The ion channel current is not smooth, it is
fractal (bursts within bursts) and therefore not
differentiable. Thus the assumptions of the
theroem are not met and we are not guaranteed
that the fractal dimension of the lag plot is
equal to the fractal dimension of the real phase
space set.
128
For example
Osborne Provenzale 1989 Physica D35381 They
used a Fourier series to generate a fractal time
series whose power spectra was 1/f . They
randomized the phases of the terms in the Fourier
series so that the fractal dimension of the real
phase space set was infinite. But, they found
that the fractal dimension of the lag plots was
as low as 1.
129
Pathological example where an infinite
dimensional random process has a LOW dimension
attractor
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
RANDOMLY pick numbers
130
Pathological example where an infinite
dimensional random process has a LOW dimension
attractor
Time Series 6, 6, 6, 6, 6, 6, 6, 6 ... Phase
Space
D 0
6
6
6
131
Organization of the Vectors in the Phase Space
Set Kaplan and Glass 1992 Phys. Rev. Lett.
68427-430
Random
no uniform flow
small
average direction
132
Organization of the Vectors in the Phase Space
Set Kaplan and Glass 1992 Phys. Rev. Lett.
68427-430
Deterministic
uniform flow
large
average direction
133
Surrogate Data Set Theiler et al. 1992 Physica
D5877-94
original time series
surrogate time series
same first order correlations higher orders
scrambled
R A N D O M
original phase space set
surrogate phase space set
same
134
Surrogate Data Set Theiler et al. 1992 Physica
D5877-94
surrogate time series
original time series
DETERMINISTIC
same first order correlations higher orders
scrambled
surrogate phase space set
original phase space set
different
135
Experiments
WEAK
Time Series phase space
Dimension Low deterministic High random
examples ECG, EEG
136
Experiments
STRONG
vary a parameter
see behavior
predicted by a nonlinear model
electrical stimulation of cells, biochemical
reactions
examples
137
Control
Non-Chaotic System
system output
control parameter
138
Control
Chaotic System
system output
control parameter
139
Control of Chaos light intensity of a laser Roy
et al. 1992 Phys. Rev. Lett. 681259-1262
NO CONTROL
Intensity
0
0.5 msec
140
Control of Chaos light intensity of a laser Roy
et al. 1992 Phys. Rev. Lett. 681259-1262
CONTROL
Control
Intensity
0
0.2 msec
141
Control of Chaos light intensity of a laser Roy
et al. 1992 Phys. Rev. Lett. 681259-1262
CONTROL
Control
Intensity
0
0.2 msec
142
Control of Chaos motion of a magnetoelastic
ribbon Ditto, Rauseo, and Spano 1990 Phys. Rev.
Lett. 653211-3214
B 0
electromagnets
magnetoelastic ribbon
143
Control of Chaos motion of a magnetoelastic
ribbon Ditto, Rauseo, and Spano 1990 Phys. Rev.
Lett. 653211-3214
B gt B1
144
Control of Chaos motion of a magnetoelastic
ribbon Ditto, Rauseo, and Spano 1990 Phys. Rev.
Lett. 653211-3214
2 T
B Bo sin ( t)
X
Xn X (t nT)
sensor
145
Control of Chaos motion of a magnetoelastic
ribbon Ditto, Rauseo, and Spano 1990 Phys. Rev.
Lett. 653211-3214
iteration number
control
0 - 2359 2360 - 4799 4800 - 7099 7100 - 10000
none period 1 period 2 period 1
146
Control of Chaos motion of a magnetoelastic
ribbon Ditto, Rauseo, and Spano 1990 Phys. Rev.
Lett. 653211-3214
4.5
4.0
Xn
3.5
3.0
2.5
2000
4000
6000
8000
10000
0
Iteration Number
147
Control of Biological Systems
The Old Way Brute Force Control. BIG machine BIG
power
Amps
Heart
148
Control of Biological Systems
The New Way Cleverly timed, delicate
pulses. little machine little power
mA
Heart
149
How do we think of biological systems?
The Old Way Forces drive the system between
stable states.
150
How do we think of biological systems?
Stable State C
Stable State A
Force D
Force E
Stable State B
151
How do we think of biological systems?
The New Way Hanging around for a while in one
condition forces the system into another
condition.
152
How do we think of biological systems?
Unstable State C
Unstable State A
Dynamics of A
Dynamics of B
Unstable State B
153
Summary of Chaos
FEW INDEPENDENT VARIABLES Behavior is so complex
that it mimics random behavior.
154
Summary of Chaos
DYNAMICAL SYSTEM DETERMINISTIC
The value of the variables at the next instant
in time can be calculated from their values at
the previous instant in time. xi (t t) f (xi
(t))
155
Summary of Chaos
SENSITIVITY TO INITIAL CONDITIONS NOT PREDICTABLE
IN THE LONG RUN
x1(t t) - x2(t t) Ae t
156
Summary of Chaos
STRANGE ATTRACTOR Phase space is low
dimensional (often fractal).
157
Books About Chaos
introductory
J. Gleick Chaos Making a New Science
1987 Viking
158
Books About Chaos
intermediate mathematics
F. C. Moon Chaotic and Fractal Dynamics
1992 John Wiley Sons
159
Books About Chaos
advanced mathematics
J. Guckenheimer P. Holmes Nonlinear
Oscillations, Dynamical Systems, and
Bifurcations of Vector Fields 1983
Springer-Verlag E. Ott Chaos in Dynamical
Systems 1993 Cambridge Univ. Press
160
Books About Chaos
reviews of chaos in biology
A. V. Holden Chaos 1986 Princeton Univ.
Press E. L. Moskilde Complexity, Chaos
and Biological Evolution 1991 Plenum
161
Books About Chaos
reviews of chaos in biology
J. Bassingthwaighte, L. Liebovitch, B. West
Fractal Physiology 1994 Oxford Univ.
Press