Introduction to CHAOS
Larry Liebovitch, Ph.D.
Florida Atlantic University
2004
These two sets of data have the same
Data 1
RANDOM
random
x(n) = RND
CHAOS
Deterministic
x(n+1) = 3.95 x(n) [1-x(n)]
Data 2
etc.
Data 1
RANDOM
random
x(n) = RND
Data 2
CHAOS
deterministic
x(n+1) = 3.95 x(n) [1-x(n)]
x(n+1)
x(n)
Definition
CHAOS
Deterministic
predict that value
these values
CHAOS
Small Number of Variables
x(n+1) = f(x(n), x(n-1), x(n-2))
Definition
Definition
CHAOS
Complex Output
Properties
CHAOS
Phase Space is Low Dimensional
phase space
d , random
d = 1, chaos
Properties
CHAOS
Sensitivity to Initial Conditions
nearly identical
initial values
very different
final values
Properties
CHAOS
Bifurcations
small change
in a parameter
one pattern
another pattern
Time Series
X(t)
Y(t)
Z(t)
embedding
Phase Space
X(t)
Z(t)
phase
space set
Y(t)
Attractors in Phase Space
Logistic Equation
X(n+1)
X(n)
X(n+1) = 3.95 X(n) [1-X(n)]
Attractors in Phase Space
Lorenz Equations
X(t)
Z(t)
Y(t)
X(n+1)
X(n)
Logistic Equation
phase space
time series
d<1
The number of independent variables = smallest integer >
the fractal dimension of the attractor
d < 1, therefore, the equation of the time series that produced this attractor depends on 1 independent variable.
Lorenz Equations
phase space
time series
d =2.03
The number of independent variables = smallest integer >
the fractal dimension of the attractor
d = 2.03, therefore, the equation of the time series that produced this attractor depends on 3 independent variables.
X(t)
Z(t)
Y(t)
X(n+1)
n
Data 1
time series
phase space
d
Since ,
the time series
was produced
by a random
mechanism.
d
Data 2
time series
phase space
d = 1
Since d = 1,
the time series
was produced by
a deterministic
mechanism.
Constructed by direct measurement:
Phase Space
Each point in the phase space set has coordinates
X(t), Y(t), Z(t)
Measure X(t), Y(t), Z(t)
Z(t)
X(t)
Y(t)
Constructed from one variable
Phase Space
Takens’ Theorem
Takens 1981 In Dynamical Systems
and Turbulence Ed. Rand & Young,
Springer-Verlag, pp. 366 - 381
X(t+ t)
X(t+2 t)
X(t)
Each point in the
phase space set
has coordinates
X(t), X(t + t), X(t+2 t)
velocity (cm/sec)
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
-10-1
-10-4
3 x 10-5
displacement (cm)
stimulus = 171 Hz
velocity (cm/sec)
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
5 x 10-6
displacement (cm)
stimulus = 610 Hz
-3 x 10-2
3 x 10-2
-2 x 10-5
Data 1
RANDOM
x(n) = RND
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
Data 2
CHAOS
deterministic
x(n+1) = 3.95 x(n) [1 - x(n)]
fractal dimension
of phase space set
fractal demension of the phase space set = 1
embedding dimension = number of values of the data taken at a time to produce the phase space set
microelectrode
chick heart cell
current
source
voltmeter
Chick Heart Cells
v
Glass, Guevara, Bélair & Shrier.
1984 Phys. Rev. A29:1348 - 1357
Spontaneous Beating,
No External Stlimulation
Chick Heart Cells
voltage
time
Periodically Stimulated
2 stimulations - 1 beat
Chick Heart Cells
2:1
Chick Heart Cells
1:1
Periodically Stimulated
1 stimulation - 1 beat
Chick Heart Cells
2:3
Periodically Stimulated
2 stimulations - 3 beats
periodic stimulation - chaotic response
The Pattern of Beating
of Chick Heart Cells
Glass, Guevara, Bélair & Shrier.1984 Phys. Rev. A29:1348 - 1357
= phase of the beat with respect to the stimulus
The Pattern of Beating of Chick Heart Cells continued
phase vs. previous phase
0.5
0
0.5
1.0
1.0
0
0.5
1.0
i + 1
experiment
i
theory (circle map)
The Pattern of Beating
of Chick Heart Cells
Glass, Guevara, Belair & Shrier.1984 Phys. Rev. A29:1348 - 1357
Since the phase space set is
1-dimensional, the timing
between the beats of these
cells can be described by a
deterministic relationship.
Procedure
e.g. voltage as a function of time
This is called embedding.
Procedure
Especially, the fractal dimension.
= Random = chance
Low Fractal Dimension
= Chaos = deterministic
The Fractal Dimension �is NOT equal to �The Fractal Dimension
Fractal Dimension:�How many new pieces of the Time Series are found when viewed at finer time resolution.
X
time
d
Fractal Dimension:�The Dimension of the Attractor in �Phase Space is related to the�Number of Independent Variables.
X
time
d
x(t)
x(t+ t)
x(t+2 t)
Mechanism that Generated the Data
Chance
d(phase space set)
Determinism
d(phase space set)
= low
Data
x(t)
t
?
Lorenz�1963 J. Atmos. Sci. 20:13-141
C O L D
Model
HOT
(Rayleigh, Saltzman)
Lorenz�1963 J. Atmos. Sci. 20:13-141
Equations
Lorenz�1963 J. Atmos. Sci. 20:13-141
X = speed of the convective circulation X > 0 clockwise, X < 0 counterclockwise
Y = temperature difference between rising and falling fluid
Equations
Lorenz�1963 J. Atmos. Sci. 20:13-141
Z = bottom to top temperature minus the linear gradient
Equations
Lorenz�1963 J. Atmos. Sci. 20:13-141
Phase Space
Z
X
Y
Lorenz Attractor
X < 0
X > 0
cylinder of air rotating counter-clockwise
cylinder of air rotating clockwise
Sensitivity to Initial Conditions�Lorenz Equations
IXtop(t) - Xbottom(t)I e t
= Liapunov Exponent
X(t)
X= 1.00001
Initial Condition:
different
same
X(t)
X= 1.
0
0
Deterministic, Non-Chaotic
X(n+1) = f {X(n)}
Accuracy of values
computed for X(n):
1.736 2.345 3.254
5.455 4.876 4.234
3.212
Deterministic, Chaotic
X(n+1) = f {X(n)}
Accuracy of values
computed for X(n):
3.455 3.45? 3.4??
3.??? ? ? ?
Initial Conditions� X(t0), Y(t0), Z(t0)...
Clockwork Universe
determimistic non-chaotic
Can
compute
all future
X(t), Y(t), Z(t)...
Equations
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
Lorenz Strange Attractor
Trajectories from outside:
pulled TOWARDS it
why its called an attractor
starting away:
Lorenz Strange Attractor
Trajectories on the attractor:
pushed APART from each other
sensitivity to initial conditions
starting on:
“Strange”�attractor is fractal
phase space set
not strange
strange
“Chaotic”�sensitivity to initial conditions
time series
not chaotic
chaotic
X(t)
t
X(t)
t
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
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. We’re on an exact trajectory, just not on the one we thought we were on.
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 actually
compute.
Trajectory that we are trying to compute.
Sensitivity to initial conditions means that the conditions of an experiment can be quite similar, but that the results can be quite different.
+
TUESDAY
10 µl
ArT
+
10 µl
WEDNESDAY
ArT
A = 3.22
X(n)
n
X(n + 1) = A X(n) [1 -X (n)]
A = 3.42
X(n)
n
X(n + 1) = A X(n) [1 -X (n)]
A = 3.62
X(n)
n
Bifurcation
compute x(2) from x(1).
compute x(3) from x(2) and
so on... up to x(300).
x(n + 1) = A x(n) [1 -x(n)]
are the transient values off
of the attractor.
Y-axis over the value of A
on the X-axis.
repeat the procedure again.
x(n + 1) = A x(n) [1 -x(n)]
Sudden changes of the pattern indicate bifurcations ( )
x(n)
x(n)
The energy in glucose is transfered to ATP. ATP is used as an energy source to drive biochemical reactions.
Glycolysis
+
-
-
periodic
Theory
Markus and Hess 1985 Arch. Biol. Med. Exp. 18:261-271
Glycolysis
time
sugar input
ATP output
chaotic
time
time
time
Experiments
Hess and Markus 1987 Trends. Biomed. Sci. 12:45-48
cell-free extracts from baker’s yeast
Glycolysis
ATP measured by fluorescence glucose input
time
Experiments
Hess and Markus 1987 Trends. Biomed. Sci. 12:45-48
Periodic
fluorescence
Glycolysis
Vin
Glycolysis
Experiments
Hess and Markus 1987 Trends. Biomed. Sci. 12:45-48
Chaotic
20 min
Glycolysis
Markus et al. 1985. Biophys. Chem 22:95-105
Bifurcation Diagram
chaos
theory
experiment
Glycolysis
Markus et al. 1985. Biophys. Chem 22:95-105
ADP measured at
the same phase each time of the input sugar flow cycle
(ATP is related to ADP)
period of the input sugar flow cycle
# =
period of the ATP
concentration
frequency of the input sugar flow cycle
Phase Transitions
Haken 1983 Synergetics: An Introduction Springer-Verlag
Kelso 1995 Dynamic Patterns MIT Press
Tap the left index finger
in-phase with the tick
of the metronome.
Try to tap the right index finger out-of-phase with the tick of the metronome.
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.
Position of Right Index Finger
Position of Left Index Finger
A. TIME SERIES
Phase Transitions
Haken 1983 Synergetics: An Introduction Springer-Verlag
Kelso 1995 Dynamic Patterns MIT Press
ADD
ABD
Position of Right Index Finger
360o
0o
B. POINT ESTIMATE OF RELATIVE PHASE
180o
Self-Organized Phase Transitions
Haken 1983 Synergetics: An Introduction Springer-Verlag
Kelso 1995 Dynamic Patterns MIT Press
2 sec
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
Phase Transition
Haken 1983 Synergetics: An Introduction Springer-Verlag
Kelso 1995 Dynamic Patterns MIT Press
+
Small changes in parameters can produce large changes in behavior.
+
10cc ArT
9cc ArT
Bifurcations can be used to test if a system is deterministic.
Deterministic
Mathematical Model
Experiment
observed bifurcations
predicted bifurcations
Match ?
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
X(t+ t)
Phase Space
Set
X(t)
The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.
Mechanism that generated the experimental data.
Deterministic
Random
d = low
d
The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.
Epidemics
Schaffer and Kot 1986 Chaos ed. Holden,
Princeton Univ. Press
4000
15000
0
0
measles
New York
time series:
phase space:
chickenpox
Epidemics
Olsen and Schaffer 1990 Science 249:499-504
dimension of attractor in phase space
measles
chickenpox
Kobenhavn 3.1 3.4
Milwaukee 2.6 3.2
St. Louis 2.2 2.7
New York 2.7 3.3
Epidemics
Olsen and Schaffer 1990 Science 249:499-504
SEIR models - 4 independent
variables
S susceptible
E exposed, but not yet
infectious
I infectious
R recovered
Epidemics
Olsen and Schaffer 1990 Science 249:499-504
Conclusion:
measles: chaotic
chickenpox: noisy yearly cycle
time series: voltage
Kaplan and Cohen 1990 Circ. Res. 67:886-892
normal
fibrillation
death
D = 1
chaos
D =
random
Phase space
V(t), V(t+ t)
Electrocardiogram
ECG: Electrical recording of the muscle activity of the heart.
8
time series: voltage
Babloyantz and Destexhe 1988 Biol. Cybern. 58:203-211
normal
D = 6
chaos
Electrocardiogram
ECG: Electrical recording of the muscle activity of the heart.
Electrocardiogram
ECG: Electrical recording of the muscle activity of the heart.
time series: time between heartbeats
Babloyantz and Destexhe 1988 Biol. Cybern. 58:203-211
normal
D = 6
chaos
fibrillation death
D = 4
chaos
induced arrhythmias
D = 3
chaos
Evans, Khan, Garfinkel, Kass, Albano, and Diamond 1989 Circ. Suppl. 80:II-134
Zbilut, Mayer-Kress, Sobotka, O’Toole and Thomas 1989 Biol. Cybern, 61:371-381
Electroencephalogram
EEG: Electrical recording of the nerve activity of the brain.
Mayer-Kress and Layne 1987 Ann. N.Y. Acad. Sci. 504:62-78
time series: V(t)
phase space:
D=8 chaos
V(t)
V(t+ t)
Rapp, Bashore, Martinerie, Albano, Zimmerman, and Mees 1989 Brain Topography 2:99-118
Babloyantz and Destexhe 1988 In: From Chemical to Biological Organization ed. Markus, Muller, and Nicolis, Springer-Verlag
Xu and Xu 1988 Bull. Math. Biol. 5:559-565
Electroencephalogram
EEG: Electrical recording of the nerve activity of the brain.
Different groups find different dimensions under the same experimental conditions.
Electroencephalogram
EEG: Electrical recording of the nerve activity of the brain.
mental task
quiet awake, eyes closed
quiet sleep
brain virus: Creutzfeld-
Jakob
Epilepsy: petit mal
meditation: Qi-kong
Electroencephalogram
EEG: Electrical recording of the nerve activity of the brain.
perhaps:
High Dimension
Low Dimension
Random Markov
How to compute the next x(n):
Each t pick a random
number 0 < R < 1
If open, and R < pc, then close.
If closed, and R < po, then open.
Random Markov
t
closed
If closed:
probability to open in the
next t=po
If open:
probability to close in the next t = pc
open
Deterministic Iterated Map
Liebovitch & Tóth 1991 J. Theor. Biol. 148:243-267
x(n) = the current at time n
x(n+1) = f (x(n))
open
closed
x(n+1)
x(n)
0
x(1)
0
x(2)
0
x(3)
0
x(2)
Deterministic Iterated Map
Liebovitch & Tóth 1991 J. Theor. Biol. 148:243-267
How to compute the next x(n):
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. 59:118-124
Tacoma Narrows Bridge
Equation of simple, forced resonance:
x + Ax + Bx = f ( t )
Equation of flutter that destroyed
the Tacoma Narrows Bridge:
x + Ax + Bx = f ( x, x )
Tacoma Narrows Bridge
Scanlan and Vellozzi 1980 in Long Span Bridges ed. Cohen and Birdsall pp. 247-263 NYAS
Wind Tunnel Tests
AIRFOIL
ORIGIONAL
TACOMA NARROWS
(
Tacoma Narrows Bridge
The drag on an airplane wing (A) increases with wind speed.
Wind Tunnel Tests
0.3
0.2
0.1
0
0.1
0.2
A2
OTN
U
NB
A
*
The drag on the OTN (original Tacoma Narrows) bridge changes sign as the wind speed increases, it enters into positive feedback.
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.
CLOSED
random
OPEN
energy
Random
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.
CLOSED
OPEN
energy
deterministic
Analyzing Experimental Data
In principle, you can tell if the
data was generated by a random
or a deterministic mechanism.
The Good News:
Analyzing Experimental Data
In practice, it isn’t easy.
The Bad News:
Need Lots of Data
attractor evenly.
Sample too often: only see 1-d trajectories.
Sample too rarely; don’t see the attractor at all.
Why it’s Hard to Tell Random from Deterministic Mechanisms
Why it’s Hard to Tell Random from Deterministic Mechanisms
Analyzing the Data is Tricky
Mathematics is Not Known
Why it’s Hard to Tell Random from Deterministic Mechanisms
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
How Many Times Series Values?
Smith 1988
Phys. Lett. A133:283 42D
5,000,000,000
Wolff et al. 1985
Physica D16:285 30D
700,000,000
Wolf et al. 1985
Physica D16:285 10D
1,000,000
How Many Times Series Values?
Nerenberg &
Essex 1990
Phys. Rev.
A42:7065
D+2
2
_______1________
kd1/2[A In (k)](D+2)/2
D/2
2(k-1) ((D+4)/2)
(1/2) ((D+3)/2)
x[
]
200,000
How Many Times Series Values?
Ding et al. 1993
Phys. Rev.
Lett. 70:3872
10D/2
(D/2)!
D/2
10
1,000
Gershenfeld
1990 preprint
2D
Lorenz
t 0
X(t+ t)
X(t)
Lorenz
X(t+ t)
X(t)
t Just Right
t correlation time
Lorenz
X(t+ t)
X(t)
8
t
Takens’ Theorem
If
X(t+ t)
X(t)
Then,
the lag plot constructed from the data
dX(t)
dt
X(t)
Is a linear transformation
of the real phase space
dX(t)
dt
because
X(t+ t) - X(t)
t
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.
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.
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.
For example:
Osborne & Provenzale 1989
Physica D35:381
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.
For example:
RANDOMLY pick numbers
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
Time Series: 6, 6, 6, 6, 6, 6, 6, 6 ...
Phase Space:
Pathological example where an infinite dimensional random process has a LOW dimension attractor
D = 0
6
6
6
Organization of the Vectors in the Phase Space Set
Kaplan and Glass 1992 Phys. Rev. Lett. 68:427-430
small
average direction
Random
no uniform flow
Organization of the Vectors in the Phase Space Set
Kaplan and Glass 1992 Phys. Rev. Lett. 68:427-430
large
Deterministic
average direction
uniform flow
Surrogate Data Set
Theiler et al. 1992 Physica D58:77-94
original phase space set
surrogate phase space set
same
original time
series
surrogate time series
R
A
N
D
O
M
same first order
correlations
higher orders
scrambled
Surrogate Data Set
Theiler et al. 1992 Physica D58:77-94
surrogate phase space set
different
surrogate time series
same first order
correlations
higher orders
scrambled
DETERMINISTIC
original phase space set
original time
series
Time Series phase space
Experiments
Dimension
Low = deterministic
High = random
examples: ECG, EEG
WEAK
vary a parameter
Experiments
predicted by a nonlinear model
STRONG
see behavior
electrical stimulation
of cells, biochemical reactions
examples:
Control
system output
Non-Chaotic System
control parameter
Control
system output
Chaotic System
control parameter
Control of Chaos
light intensity of a laser
Roy et al. 1992 Phys. Rev. Lett. 68:1259-1262
NO CONTROL
0
0.5 msec
Intensity
Control of Chaos
light intensity of a laser
Roy et al. 1992 Phys. Rev. Lett. 68:1259-1262
CONTROL
0
0.2 msec
Intensity
Control
Control of Chaos
light intensity of a laser
Roy et al. 1992 Phys. Rev. Lett. 68:1259-1262
CONTROL
0
0.2 msec
Intensity
Control
Control of Chaos
motion of a magnetoelastic ribbon
Ditto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214
electromagnets
magnetoelastic
ribbon
B = 0
B > B1
Control of Chaos
motion of a magnetoelastic ribbon
Ditto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214
Control of Chaos
motion of a magnetoelastic ribbon
Ditto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214
sensor
X
Xn = X (t = nT)
2
T
B = Bo sin ( t)
Control of Chaos
motion of a magnetoelastic ribbon
Ditto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214
iteration
number
0 - 2359
2360 - 4799
4800 - 7099
7100 - 10000
none
period 1
period 2
period 1
control
Control of Chaos
motion of a magnetoelastic ribbon
Ditto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214
4.5
4.0
3.5
3.0
2.5
0
2000
4000
6000
8000
10000
Iteration Number
Xn
Control of Biological Systems
The Old Way
Brute Force Control.
BIG machine
BIG power
Heart
Amps
Control of Biological Systems
The New Way
Cleverly timed, delicate pulses.
little machine
little power
mA
Heart
The Old Way
Forces drive the system between stable states.
How do we think of biological systems?
How do we think of biological systems?
Force D
Force E
Stable State B
Stable State A
Stable State C
How do we think of biological systems?
The New Way
Hanging around for a while in one condition forces the system into another condition.
Dynamics of A
Dynamics of B
How do we think of biological systems?
Unstable State B
Unstable State A
Unstable State C
Summary of Chaos
FEW INDEPENDENT VARIABLES
Behavior is so complex that it mimics random behavior.
Summary of Chaos
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))
DYNAMICAL SYSTEM
DETERMINISTIC
Summary of Chaos
x1(t+ t) - x2(t+ t) = Ae t
SENSITIVITY TO INITIAL CONDITIONS
NOT PREDICTABLE IN THE LONG RUN
Summary of Chaos
STRANGE ATTRACTOR
Phase space is low dimensional (often fractal).
Books About Chaos
J. Gleick
Chaos: Making a New Science
1987 Viking
introductory
Books About Chaos
F. C. Moon
Chaotic and Fractal Dynamics
1992 John Wiley & Sons
intermediate mathematics
Books About Chaos
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
advanced mathematics
A. V. Holden
Chaos
1986 Princeton Univ. Press
E. & L. Moskilde
Complexity, Chaos and
Biological Evolution
1991 Plenum
reviews of chaos in biology
Books About Chaos
Books About Chaos
J. Bassingthwaighte,
L. Liebovitch, & B. West
Fractal Physiology
1994 Oxford Univ. Press
reviews of chaos in biology