CARESSWorking Paper #02-02
"CHAOS THEORY AND ITS APPLICATION"
By
Hairn H. Ban and Yochanan Shachrnurove
~
UNIVERSITY of PENNSYL VANIA
Centerfor Analytic Research
in Economics and the Social Sciences'
McNEILB~PINO,3.718 LOCUSTW~g
PHll..ADELPHlA, PA 19104-6297
Chaos
Theory
and Its
Application
Haim H. Bau
Department
University
of Mechanical
of
Engineering
Pennsylvania,
E-mail:
and Applied
Philadelphia,
Mechanics,
PA 19104-6315,
USA
bau@eniac.seas.upenn.edu
And
Yochanan Shachmurove
Departments
of Economics,
The City
College
of New York and The University
E-mail:
of The City
of Pennsylvania
Yochanan@econ.sas.upenn.edu
University
1;
Abstract
One consequence
simple
systems,
equations,
reveal
which
can
behavior.
strategies
into
conditions
systems
are
computational
described
be
very
are
are
by
processes
underlying
is
that
relatively
a
few
non-linear
stochastic-like
inexpensively.
mechanisms
They
while
devising
processes.
sensitive
not
to
initial
precisely
long-term
predictions
impossible.
Thus,
for
theory
complicated,
simulate
resources
predictions
can
these
systems
perturbations,
system
the
to control
Chaotic
dynamic
exhibit
Such models
insights
these
of
will
not
systems
known
of
the
enable
ranging
conditions.
and
the
are
subject
behavior
availability
weather
of
of
one to generate
from
Since
to
these
large
long-term
to
economic
forecasts.
Key Words:
Behavior,
Chaos
Lyapunov
Random-looking
Stirrer,
Information
Phase
Theory,
Deterministic
Exponent,
Non-linear
Models,
Poinare,
Magneto
Oscillations,
Space,
Dimension.
Embedding
Systems,
Theorem,
Stochastic
Lorenz
Loop,
Hydrodynamic
Power
Spectrum,
~~
Introduction
The
last
few
mathematical
equations:
of
modeling.
the
various
either
of
great
in
the
the
for
oscillations
population
of
the
classified
as
either
intention
is
to
speaking,
mathematical
when repeated
exactly
the
we define
available,
the
of
such models
system,
streams
rhythms,
models
to
Here,
and
can be
Since
our
make
the
process
as
same way will
stochastic
when repeated.
to
conclude
behavior
one should
and that
be able
behavior.
In other
words,
we should
be
to
able
in
have
a
yield
processes
will
we will
focus
processes.
One may be tempted
is
deterministic
In contrast,
outcomes
regular
a
and
or
sciences,
stochastic.
jargon
exactly
same outcome.
only
or
technical
that
on deterministic
weather
Broadly
process
solely
a few examples
deterministic
processes
natural
physiological
simple,
different
and
the
written
Such models
diseases,
avoid
yield
statements
of
presentation
exhibit
Just
of evolution
continuous
physicaJ
on
or the dynamics
processes.
a pendulum,
spread
dynamics.
for
emphasis
consists
mathematical
the
and economics.
increased
time-dependence
discrete
in
an
of models
equations
utility
ocean,
seen
of the
through
equations
engineering,
are
description
differential
difference
have
One class
processes
as
been
decades
tell
if
to
that
systems
once a deterministic
predict
we know the
the
deterministic
system's
the
model
system's
future
system's
current
state,
future
states
at
all
4
times.
Although
many systems
there
are many others
behavior,
many deterministic
behavior.
systems
Such
reemphasize
systems
in
of
the
of this
paper,
systems
that
characteristics
conditions.
sensitivity
to
initial
conditions,
Any small
system,
inaccuracy
measurement
long-term
errors,
prediction
growing
rapidly
systems
included)
known is
the
exhibited
that
fact
amplify
rapidly
that
exhibit
"...
Henri
it
conditions
error
in
the
is
small
will
Prediction
becomes
fortuitous
phenomenon"
(Poincare,
the
complex
behavior
high
behavior.
such as may result
and
will
of
unbounded
to
credited
latter.
of
high
exhibits
future
from
render
any
initial
errors
systems
(linear
growth.
Less
initial
mathematician,
great
former
is
widely
conditions
is
bounded systems.
French
very
its
exponential
may happen that
produce
all
such sensitivity
Poincare
behavior.
systems
The possibility
common to
prominent
astronomer,
small
is
predict
data,
useless.
complex
chaotic
even when we have an accurate
initial
by many nonlinear,
The
that
will
We
we define
chaotic
are
chaotic.
When a system
we cannot
in the
there
random-like
as
exhibit
of
fact,
irregular,
the context
initial
that
exhibit
and predictive
In
to
to
for
do not.
referred
sensitivity
model
regular
are
as deterministic
One
that
that
systems
that,
do exhibit
exhibited
ones
dynamist,
to be the
differences
in
produce
the
first
in
final
1913).
by chaotic
and
to realize
the
initial
phenomena.
an enormous
impossible,
and
error
we
in
have
A
the
the
A detailed
depiction
systems
was delayed
I~(
until
the
appearance
numerically
long
the
time
1963,
of computers
behavior
intervals
while
weather
of
and discrete
the
simple-looking
differential
equations
system
recognized
community
systems
may
exhibit
random-like,
dogma had been that
demonstrated
degrees
with
that
of
very
a
freedom
behavior.
Although
behavior
that
systems,
such
dynamics,
the
irregularity
and not
The
as three)
fact
that
behavior
results
from
that
a
behavior.
deterministic
would
freedom.
the
be exhibited
Lorenz's
small
work
number
exhibit
may exhibit
of
the
behavior,
may also
systems
for
nonlinear
(chaotic)
a relatively
deterministic
resembles
of
In
observed
coupled,
such behavior
with
few
models
turbulent
many degrees
system
(as
three
complex
scientific
by systems
of
relatively
many iterations.
Lorenz
exhibits
the
only
N.
investigate
over
simplified
E.
Although
prevailing
over
various
one to
models
models
meteorologist
deceptively
allowed
continuous
he was studying
system,
that
of
chaotic
random-like
stochastic
the
system's
may
exhibit
(random)
intrinsic
random influences.
that
certain
systems
chaotic,
..
unpredictable
behavior
philosophical
chaotic
implications.
systems
computational
acquire,
the
the
cannot
power.
chaotic
long-term.
exhibit
has
chaotic,
very
The
be
cured
No matter
system
by
lack
of
increases
how large
will
The realization
complex
important
behavior
still
that
practical
and
predictability
of
in
a computer
remain
that
and
one would
unpredictable
low-dimension
suggests
computer
systems
over
may
some complex
"
j6
phenomena
that
appear
describable
by
One of the
targets
or no success)
assurance
the
short-term
states,
of
chaotic
systems
behavior
many
of
intervention.
induce
For
of
as well
the
last
not
so excellent
professional
new book
or
to
books
is
as chemical
compendium
it
may be
are a great
on this
A month will
devoted
to
not
to
occur.
with
high
for
reactions.
of
number of
chaos
minimal
desirable
crosses
as well
hardly
with
beneficial
is
amount
topic
chaotic
extracting
with
normally
and biological
a great
many
their
associated
which
some of
Finally,
of
often
can
estimator
opportunity
phenomenon that
focused
a state
system
mixing
There
observes
single
no
chaotic
estimates
periodically
would
is
long-term
behave
when it
behavior
journals.
to
the
a
has attracted
two decades.
that
devise
occasionally
a ubiquitous
The topic
them
from
and
error
One can suppress
conditions
stirring
Chaos is
of
leads
chaotic
homogenization
lines.
This
behavior
under
example,
levels
induce
Moreover,
chaos
to
defy
predictions.
are controllable.
periods.
types
possible
or
from
dynamics
an observer
little
there
result
be
models.
with
Of course,
with
and modifying
altogether
various
is
may
mathematical
market.
chaotic
with
sight
(apparently
predictions
it
updating
first
fluctuations
the
Furthermore,
capable
stock
market
predictions,
system's
at
low-dimensional
has been the
Although
the
random
of such investigations
dynamics.
made.
be
relatively
that
still
to
disciplinary
attention
excellent
and
as a number
go by without
theory
in
appearing
a
in
print.
of
There
research
are
on this
highly
readable
from
(1987)
bestseller,
images,
to
seeking
offered
by
inordinate
toy
spatial
toy
-
are
toys.
a non
albeit
details.
table
and
literature,
highly
technical
than
the
topic
these
this
through
toys
the
were chosen
the phenomena described
exhibits
disciplinary
temporal
hydrodynamic
lines.
chaos
stirrer
that
without
To facilitate
Although
electro-magneto
coffee
exposure
and cross
-
hundreds
as Gleick's
greater
of expertise,
loop
(1986)
targets
introduce
generic
Lorenz
the
somewhat
therefore
such
literature
paper
technical
and
The literature
books
and Richter's
technical
in
areas
the
-
The
while
-
and
the
exhibits
chaos.
The Lorenz
Loop
The first
the
obtain
theory,
annually.
non-technical
This
of two chaotic
authors'
chaos
appear
Peitgen
non
shall
implications
second
it
to
immersion
description
to
engineering/physics
the
we
first
topic
manuscripts.
audience
its
and
the
mathematical
from the
devoted
papers
ranges
quest,
journals
as the
"toy"
Lorenz
Lorenz
is a thermal
loop
because
In
equations.
experimental
analog
into
(doughnut-shape)
a torus
See Figure
1 for
INSERT FIGURE 1
the
of
the
schematic
it
convection
can be approximately
other
Lorenz
loop.
words,
model.
and standing
depiction
this
We refer
modeled by
set-up
Imagine
a pipe
in the vertical
of the
apparatus.
is
an
bent
plane.
8
The tube is
of
the
apparatus
heating
axis
the
is
that
heating,
expands,
its
heating
rate
observed
time
with
in
with
denote
conditions
is
parallel
the
liquid
3
in
a certain
flow
positions
to
exceeds
loop.
is
the
gravity
vector.
the
lower
respect
to
across
the
the
When the
flow
of
the
difference
loop
as
Y,
and
6 and 12 o'clock
These
variables,
all
which
functions
to describe
the
The solid
loop.
rate
(X)
constant
as
at
the major
line
a function
in
of
features
Figure
time
of the
2 depicts
when the
of
flow
the
heating
We
between
positions
sufficient
in
flow.
between
are
is
irregularly
difference
of
the
apparatus
temperature
three
to
irregular
direction
temperature
The
As a result
rise.
oscillates
half
cooled.
of
value,
the
the
half
tends
rate
of
o'clock
half
with
critical
as X,
9
upper
and it
reversals
The lower
water)
symmetric
The flow
rate
and
the
are
decreases,
occasional
the
while
density
the
(i.e",
liquid
heated
and cooling
loop's
of
filled
the
as Z.
time,
are
dynamics
in
computed
flow
rate
held
is
some value.
INSERT FIGURE 2
The documentation
did
in
Figure
irregular,
about
2 is
a signal
referred
to
random-looking
an equilibrium
correspond,
of
to
of
as a time-series.
oscillations,
position.
respectively,
as a function
i.e.,
Positive
motion
in
as we
Witness
periodic
and negative
the
time
values
counterclockwise
the
motion
of
X
and
9
clockwise
directions.
and negative
peaks
etc.
This
heads
sequence
is
(N).
behavior
to
and
that
in
In
Figure
line).
This
second
mathematical
different
stay
model
initial
close
exhibit
to
as
high
not
continue
each other,
to
certain
values.
bounded
behavior
with
disturbances.
high
Chaotic
systems.
the
one,
that
sequence
in
of
Figure
2
about
it.
in
one can think
of
around
a mean value
species.
a second
time-series
first
one,
albeit
Although
by
with
the two signals
the
time
behaviors.
as the
system
Only
non-linear
sensitivity
is
(dashed
the
is
be
same
slightly
initially
diverge
and
a result
of
The divergence
to
can
series
This
conditions.
behavior
above
one depicted
generated
initial
indefinitely
like
was
eventually
of
2 has been observed
fluctuations
different
sensitivity
Figure
series
conditions.
exceeds
nonlinear
the
the
random
For example,
also
the
sequence
described
nothing
interacting
time
significantly
the
is
in
the
we depicted
words,
random
signal
there
many systems.
of three
2,
positive
by a positive
the
behaviors
X, Y, and Z as representing
of the populations
the
and counting
the
depicted
Furthermore,
2 prevail
of
a coin
Yet,
In other
followed
reminiscent
deterministic,
experiments.
Figure
peaks
when tossing
(P) and tails
Similar
respectively,
P and N and document the succession
two negative
obtain
fully
us denote,
as N2PN2PN2PNPN2p8
implies
one would
in
peaks with
and valleys
sequence
is
Let
does
bounded,
and X never
systems
can
initial
exhibit
conditions
exhibited
only
or
by
."
JQ
The
structure.
such
time
Indeed,
as
order,
has
we will
coordinates
in
system's
Z
of
of
analyzing
frequency
to
in
is
phase
of
(Fourier
signal
lacking
roaming
depicted
in
this
spanned
by
the
The state
of
the
system
It
series
domain
by a point
the
lack
To unravel
space
specified
space.
as a particle
a
to
time
signal
it.
space).
evolution
in
the
order
phase
time
2 seems
a broadband
signal
(the
The time
state
Figure
Nevertheless,
in
(trajectory)
the
reveal
the
and
any instant
phase space.
a curve
the
Y,
in
amount
depict
X,
at
data
spectrum)
fair
in
methods
frequencies.
Figure
system
the
and power
any dominating
depicted
traditional
presenting
transform
this
series
is
is
useful
around
in
(X, Y, 4")
depicted
by
to
of
think
space.
INSERT FIGURE 3
Figure
Witness
of
3 depicts
that
phase portrait
to
it.
twisted
No
are,
pattern.
attractor
depicted
strange
there
matter
eventually,
surface.
is
a
the
The
called
the
feature
Figure
attractor.
system
Lorenz
the
3 appear
follow
to
which
the
trajectories
Since
the
structure
Attractors
are
is
present
for
initial
will
it
amount
tendency
system's
complicated,
loop.
has a fair
trajectories
to
3 is
the
an amazing
in Figure
an attractor.
in
is
of
what
The trajectories
attracted
as
portrait
of
Indeed,
self-organization.
similar
phase
the
structure
conditions
the
lie
a
on a
are
of
referred
onl¥
the
to
in
u
dissipative
system,
energy"
but
In
example,
our
attractor
space;
dissipate
so
its
systems
energy
the
occupies
occupied
it
i.e.,
phase
zero
dimension
numerous
suggests
generalizing
the
fractional
dimensions
-known
attractor'
The
phase
concept
periodic
usually
does not
a
unique
the
in
solution.
In
conditions
have
trajectory
in phase space.
cross.
In
two-dimensional
and the most complicated
bounded
system
chaotic
behavior,
can exhibit
continuous
(three
does
to
apply
is
discrete
for
useful
This
include
our
Pc?""
obtaining
of
trajectories
behavior
evolution
once
(which
equations
the
initial
a unique
that
trajectories
cannot
trajectories
cannot
autonomous
an autonomous,
oscillations.
systems
freedom).
systems.
will
case,
solutions
closed
a
trace
that
of
the
words,
periodic
degrees
In
very
implies
space,
behavior
three-dimensional
not
This
as
have
to
2.
system
sheet
to it,
must
approximately
of
the
phase
The
dimensions.
other
the
3.
dimension
systems),
specified,
and
a surface).
pathological
physical
present.
feature
of
example,
Barring
is
attractor
nature
For
behavior.
been
is
is
on
occur
of
as fractal
equations.
indicate
"onion"
dimension
portrait
information
differential
the
dimension
space
of
than
Hence the
(2 is
s fractal
qualitative
have
2
"potential
three-dimensional
smaller
has a sort
larger
preserve
three-dimensional,
the
be
layers.
than
is
in
must
do not
as when friction
space
dimension
the
such
volume
by the attractor
contains
that
intersect
continuous,
To exhibit
must be at
least
Such a restriction
Discrete
systems
lack
1.1
continuous
trajectories;
around.
In
systems
a
and
fact,
even
may exhibit
in
phase
one-dimensional,
chaotic
one-dimensional,
points
behavior.
discrete
space
can
nonlinear
discrete
One celebrated
chaptic
system
is
jump
example
the
of
logistic
equation.
The phase
the
space
mathematical
portrait
model
in
to
can also
can carry
out
variable,
say X, as a function
other.
be reconstructed
measurements
P teeth
Guidelines
values.
One then
extracts
by the
the points
time
comb intersects
Figure
in
all
these
in
phase
This
and when the
mathematical
the dynamic
with
is
not
a relatively
not
small
the
apriori
value
of
constructs
axis
and
variables
Xl,
coordinates
The collection
useful
and one analyzes
of
of
of
the
when
the
empirical
many degrees
When the
1
traced
a description
feasibility
each
from
curve
be the
especially
model.
a single
time
the
One
desirable
the
space.
model has very
determine
is
is
known,
to
provides
of
the
obtain
considered
space
a low dimension
system
to
phase
technique
model
with
are
2)
P-dimensional
mathematical
dynamics
range
the
(i.e.,
for
1: apart
at which
a point
to
the
series.
series
distances
for
The
One then
comb along
of
and one wishes
(t).
the
These variables
attractor.
a time
time
at
available
X21 ..., Xp.
points
based on a time
of
using
trajectories.
slides
series
the
the
and obtain
placed
are
3 was constructed
compute
attractor
a "comb" with
Figure
of
data
freedom
describing
the
dimensionality
of
known,
the practice
P, for
example
is
to start
P=3, reconstruct
1.3
the
attractor,
gradually,
compute
a function
increase
generated
DA will
longer
vary
which
the
as
with
of
freedom.
system,
systems'
systems,
by a chaotic
there
indefinitely
is
as
practical
only
technique,
no
for
one
rigorous
straightforward,
with
Another
way
documenting
designated
attractor
Its
the
surface
if
noise
of
course,
that
penetration
the
points
degrees
of
a signal
stochastic
increasing
a procedure
Using
attractors.
this
of
smooth
may sound,
may not
measured
is
All
through
applications
must be filtered
analyzing
the
systems.
technique
because
is
keep
related
no
P beyond
P
In
such
different
practical
of
on
system.
low-dimension
as this
and will
whether
DA will
is
sufficiently
significant
and
are
series
value,
test
P
DA will
time
depends
us to
Of
only,
Typically,
The value
or a stochastic
As funky
foundations.
contaminated
allows
however,
transformations.
P.
of
reconstruct
attractors,
in
number
relatively
can
a constant
longer
P increases.
P.
once P is
no
The above procedure
generated
of
When the
increases
dimension
Tpen
increase
"
(DA).
eventually,
achieve
further
the
dimension
P increases.
saturate,
fractal
estimate
be
DA as
by a chaotic
large,
these
its
and
initially
is
and calculate
signals
it
has
always
may be
out.
phase
space
of
trajectories
portrait
is
by
through
a
in phase space.
INSERT FIGURE 4
Figure
Z=Zo=constant.
4 depicts
the
penetration
Such portraits
are
points
known as
through
Poincare
the
plane
sections.
)4
At first
glance,
the
Poincare
which
would
imply
that
sheet.
In
fact,
segments,
two-dimensional
discover
that
packed
sheets.
sense
that
similar
to
th
it
each
the
one through
In
the
the
continuous
periodically
in
stroboscopic
images
state.
the
is
number
line,
of
reveals
converts
The h~
the
previous
Yn-l} - > {Xn,
the
continuous
taken
once
explained
between
when a system
section
in greater
is
will
every
the
structure
us to make a connection
Often
we
in
a
to
a
closely
magnification.
section
line
to
be self-similar
relates
Poincare
(images
This
to
previous
models.
time,
of
confined
map of the form {Xn-l'
allowing
and discrete
is
large
section
Poincare
one,
consist
when we zoom on the
Poincare
the
to
attractor
a very
a two-dimensional
effect,
the
magnification
one we saw in
in
appears
appears
additional
model to a discrete
system's
of
The structure
penetration
Yn}.
consists
section
forced
consist
period)
details
of
of
the
in
the next
predictions,
short
section.
Although
terms
predictions
divergence
as the
error
of the
presence
of
observer
possible.
exponent
estimation
the
defy
trajectories.
as a function
observer,
systems
are
Lyapunov
and the
The
chaotic
state
of
One can estimate
This
error
in
interval.
of the chaotic
disturbances
continuously
rate
estimating
Moreover,
rate
the
the
with
is
one
in
initial
or
more
of
known
prediction's
initial
the
aid
system can be estimated
and uncertainty
monitors
the
of divergence
and one can estimate
the
time
long-term
state
of
an
in the
conditions.
of
the
state
variables
or
behavior
an
is
extra
some other
estimated
term
between
the
a
X as
controller
very
of
the
application
able
to
Contreras,
the
behavior
the
electrical
in
the
rat
a
al.,
tt.ij§'l).
In
is
chaotic
the
are
in
with
use
system.
a
For
beatings
Through
mode,
a rabbit.
they
the
were
Similarly,
characteristic
tissue
have been controlled
fact,
one can obtain
chaotic.
patterns
brain
to
irregular
a feedback
nearly
periodic
possible
a single
arrhythmia
when one
non-stable,
that
use of
behavior
(i.e.,
Thus,
in
the
chaotic
period.
heart
stimuli
cardiac
chaotic
Gur)
of
With
obtain
et
determined
electrical
control
apparently
epileptic
have
difference
would
numerous
from
the
behavior
It
model plus
values.
the
one
any desired
chambers
of
suppress
periodicities.
researchers
atrial
of
controllable.
See Singer
behaviors
The system's
mathematical
function
time,
contains
various
of
can
of
line).
to stabilize
example,
its
time-independent
function
many types
a
are
one
attractor
of
of
is
systems
horizontal
chaotic
orbits
a
measurant.
measured and the predicted
and obtain
straight
the
that
controller,
completely,
the aid
(filter)
chaotic
feedback
depicts
with
actually
Finally,
state-dependent
(cf.
the
paper
aid
of
of
by Gur,
a feedback
controller.
Of equal
induce
(laminar).
Efficient
chaos
interest
in
Chaotic
stirring
is
systems
systems
is
the problem
that
usually
desirable
are
of using
naturally
exhibit
in
a controller
well
efficient
chemical
and
to
behaved
stirring.
biological
t6
reactions.
We shall
in
section
the next
encounter
the
and this
will
Hydrodynamic
with
an electrode
additional
electrodes
the
on the
cavity
Insert
parallel
to
and the
magnetic
induce,
say,
motion
in
where
B
circulation
is
a
qircular
periphery.
in
by
of
a potential
be induced
is
Two
inside
will
to
C
is
around
a weak
where A is
in
forces
the
some of
the
such
turn,
cavity.
small
particles
are
as pattern
electrodes
now
B.
in
with
pattern
positive,
current
the
liquid
across
solution
that,
in
flow
positive
the
between
electrode
is
When a potential
flow
difference
that
with
circulation
this
field
filled
A-C,
seeding
We refer
and
it
Lorenz
flow
trajectories
6.
magnetic
The interaction
results
traced
negative
will
to
5.
solution.
current
clockwise
be
Figure
and
electrodes
electric
counter
When we apply
of
its
See Figure
axis,
across
field
The
depicted
also
eccentrically
in a uniform
two electrodes.
can
particles.
around
such as saline
negative,
the
A.
cavity's
applied
between
consists
deposited
bottom.
is positioned
is
and C is
The
A and B are
solution
difference
steering
5
the
electrolyte
induce
us the opportunity
stirrer
C deposited
cavity's
Figure
The cavity
give
chaos to
Stirrer
The magneto-hydrodynamic
cavity
use of
chaos..
spatial
The Magneto
explore
a
We refer
B-C,
clockwise
to this
flow
pattern
engaging
pair
electrodes
is
the
as pattern
Figure
Figure
7 depicts
are
at
that
At moderate
values
irregular
chaos
Insert
The
spread
of
contrast
to
here
of
are
present
diverge
next
as time
the period.
7b),
sections)
period.
7a),
When the
the
tracer
of patterns
one observes
further
(Figur~
of
the
tracks
A and B.
appearance
increased,
the
7c).
7
first
the
is
characterized
that
flow
the
is
chaotic
each
goes by.
In
where
types
induced
depicted
in
chaos.
In
behavior
was
the
chaotic
is
spatial.
The
to
conditions
sensitivity
words,
their
irregular
is
known as Lagrangian
other
other,
the
behavior
of the
behavior
such as high
by
chaotic
patterns
example
here.
to
each
(Figure
cavity
phenomenon
chaos
of
When T is
entire
Witness
our
Each electrode
(Poincare
a superposition
islands.
two regular
features
particles
frequency
behavior
This
end
T (Figure
the
points.
6.
by alternately
T.
to half
images
the
nearly
of
Figure
temporal;
also
high
into
by alternating
Figure
at
is
chaotic
device
a period
equal
stroboscopic
chaotic
spreads
interval
the
6
location
a trajectory
of
a time
Insert
alterations
We operate
A-C and C-B with
engaged for
tracer's
B.
if
initial
we place
trajectories
will
two
other
tracer
markedly
To see additional
trace
the
features
evolution
of
a trace
of
the chaotic
of
dye introduced
when the
period
T is
relatively
large
7).
Figure
8a,
we place
a black
8d,
8e,
In
Figures
8b,
material
blob
after
that
flow
stretches
the
8c,
and 8f
10, 20,
blob
we will
into
(same period
depict,
30, 40,
advection,
the
as in
inside
material
lines
bounded
(confined
(Figure
Figure
the
cavity.
respectively,
and 50 time
fluid
the
periods..
same
Witness
8b)
INSERT FIGURE 8
Since
the
flow
stretching
cannot
are
forced
to
the
two
fold.
and
diffusion.
This
characteristic
present
the
Beyond Physics
So far,
of
hinted
systems.
the
chaotic
depicted
does
the
in
not
cavity),
material
interface
flows
include
stretching
dissipative
systems
lines
between
provide
Figure
continuous
the
8 is
any
for
governed
molecular
and folding
and
it
is
is
also
attractor..
and Engineering
we have discussed
engineering
about
of
chaotic
Lorenz
it
why
the
and
increases
is
and
process
of
in
this
The process
alone,
to
indefinitely
The process
stirring.
kinematics
world
continue
materials
efficient
by
is
possible
Indeed,
and
concrete
physics.
extensions
biology,
finance,
examples
Earlier
to
in
taken
the
from
paper,
the
we
biological
and
social
economics,
and
social
~'9i\
c,'
phenomena yield
appear
to
wonder
whether
complex
be
(intrinsic)
such
can
influences
in
systems,
phenomena
(described
the
of
behavior
ones displayed
been
crosses
aware
Volterra
of
chaotic
proposed
and
Perhaps
the
model of population
number of
members of
appropriate
evolution
in
growth
rate.
linear
function
above
to
a single
a certain
In
of
the
Xn.
and crowding
this
is
known
equation
in
physical
or
Although
different
(and
systems,
similar
the
to
Indeed,
the
Dynamic
catches
at
check
the
as the
discrete
is
on a hand-held
of
the
logistic
(n).
the
With
number
rate
is
the
a
limited
population.
map.
as
nearly
large,
calculator
of
modeled
represents
relatively
size
Adriatic.
Xn denote
is
growth
the
the one that
the
term
1926,
explain
generation
first
have
as
the
Let
Xn+l'
the
When Xn is
in
1<A<4),
(n+1),
small,
to
dynamics
species.
the
early
model
species
above,
dynamists
As
(O<Xo<l,
When Xn is
equation
iterate
of
fish
generation
Xn+l= AXn(l-Xn).
resources
certain
normalization
individuals
quite
predator-prey
of
the
are
population
fluctuations
describes
models)
systems.
annual
simplest
endogenous
noise).
fluctuations.
a simple
to
to
lines.
biologists
population
often
natural
deterministic
than
disciplinary
many years,
is
phenomenologically
by deterministic
System Theory
For
often
that
attributed
forces
discern)
are
It
by stochastic
driving
to
be
with
(described
more difficult
patterns
evolutions
and unpredictable.
mechanisms
these
and spatial
stochastic
exogenous
often
time
It
The
is
easy
and observe
20
the
various
Clearly,
patterns
the
increases,
size
the
independent
chaotic
of
to
initial
study,
outbreaks
Models
similar
been utilized
also
the
capital
growth,
example,
are often
flow.
Nonlinear
economic
economic
chaos.
the
It
or
is
systems
For
been
to
the
between
by
used
with
things,
the
in
This,
are
economic
and
and
chaotic,
Forsyth
does
(1994)
market
of
of
spread
of
can
quantities,
cycles,
not
the
the
that
complex
observed
necessarily
claims
in
those
sense
for
to turbulent
duplicate
to
have
phenomena such as
Business
similar
in
the
models
dynamics
unpredictability
ways
in
degrees
commodity
models
however,
financial
various
cycles.
mathematical
be
of vaccinations.
prices
in their
can
as
that
cause
high
including
species,
surprising
to
with
complicated)
various
business
likened
say,
ones used in population
to simulate
and
to
model
not
time-
periodicities
competing
perhaps
A.
As A
from
The logistic
and the effectiveness
example,
international
more
among other
qualitatively,
systems.
is
of
(O~Xn~l).
vary
various
that
more
values
patterns
with
have
relationships
behaviors
bounded
made
two
dynamics
to
is
evolution
(and
model.
population
population
conditions.
between
predator-prey
various
behavior;
enriched
interactions
for
time-periodic,
(irregular)
readily
epidemic
the
evolve
population's
to
sensitivity
success
that
imply
of
in
that
deterministic
the
1990s
is
behavior
of
remarkably
~t
similar
to the behavior
a possible
deterministic
Recognizing
data
is
view.
is
important
chaotic
From the
process
high
from
and,
of
beyond the
realm
Weak-form
empirical
merely
the
"random
walk"
theories
price
cannot
at universally
market
efficiency
theory
of
maintain
current
efficiency
dynamics.
may facilitate
The
perturbations
in
error
market
that
makes
some
margins
does not
claim
market
been
the
efficiency
cannot
cases,
are
not
subject
of
hypothesis
be predicted
information.
This
reliably
earn
available
information.
should
not
efficiency
stock
and that
price.
long
prices
and market
prices
form
has
securities
distributed,
the
which
predictions.
small
estimable
The weak-form
independently
is
models,
a model
Nevertheless,
efficiency
investors
by looking
weak-form
walk
market
future
that
to
a system
underlying
short-term
systems
within
that
of
of possibility.
and past
suggests
the
such
some cases,
chaotic
scrutiny.
that
view,
points
knowing
mathematical
of
impossible.
predictions
current
in
of
chaos in economic
of view,
understanding
predictions
short-term
suggesting
and practical
constructing
point
sensitivity
long-term
theoretical
point
in
a deeper
practical
in the l8~Q$,
of deterministic
both
theoretical
control
states
existence
may assist
provide
markets
pattern.
the
From the
would
of local
Unlike
that
the
hypothesis
abnormal
returns
However,
be confused
with
(Fama, 1970).
returns
are
sole
random walk
stock
returns
of
theories,
are
the
Random
identically
indicator
from
and
future
weak-
stochastic
22
"
given
all
prices
unobserved
are
It
information.
under
stochastic,
only
does claim
past
stock
that
and market
price
data
information.
The most
is
basic
test
autocorrelation.
returns
of
the
This
to a linear
test
regression
to the
actual
models
such as the Capital
weak-form
fits
the
model.
minus the expected
Asset
efficiency
hypothesis
time-series
The excess
of
return
excess
is
equal
determined
from
return,
which
is
Pricing
Model
(Fama and MacBeth,
1973)
Autocorrelation
market
studies
efficiency
relationship
theory.
between
fourteen
weeks
correlation.
(1965)
forty-five
finds
performed
correlational
weekly
returns
correlation,
a portfolio
of
the
previous
of
of
week.
tool
rules
stock
stocks
states
no
one and
significant
show
0.026
for
stocks.
,
portfolios
some
small
Fama
a period
(1988)
of
have
They find
show almost
of the weekly
correlation
by the
no
return
return
among small-stock
trading.
analysis
for
the
For example,
percent
from infrequent
that,
over
can be explained
The higher
of technical
returns,
returns.
on portfolio
large-cap
smaller
studies
do
of
weak-form
(1974)
and Conrad and Kaul
as much as nine
may result
Another
class
of
(1988)
studies
whereas
of
daily
the
finds
studies
coefficient
Lo and MacKinlay
portfolios
some
a correlation
stock
and
successive
one day.
that
U.S.
respectively,
between
support
Cootner
Meanwhile,
correlation
strongly
is
a given
the
filter
security,
rule.
the
This
price
~'
~ 3
fluctuates
between
new information
range
may
comes to
shift
equilibrium
gains
Besides
tools
were
test
and the
number
of
by
such
that
their
support
movements.
unpredictable.
the
weak-form
small
price
earn
is
market
since
excess
Thus,
and
profits
it
can
appears
market
data,
it
requires
the
the
small
claim
evidence
traditional
would
by
given
stock
a
efficiency.
earned
that,
run
filter
analysts
profits,
costs
be
the
transaction
measures
hypothesis.
transaction
the
useful
small
with
technical
efficiency
of
as
since
of
on
time,
consistent
preponderance
the
have studied
a very
of market
of
losses.
(1966)
abnormal
presumption
that
true
available
can
their
clear
particularly
This
new
a breakout
number
over
and
a
one
such
Although
In summary, although
support
a
unprofitable
and selling.
to
capitalize
hypothesis,
strategy
When
price
shift
further
Fama and Blume
strategy
techniques
the
whatever
a
to
to avoid
sizes.
fair
when such
stock
the
price.
through
approaches,
test
filter
the
A
that,
the
a buy-and-hold
buying
is
claim
stock
rule.
possible
correlation.
It
to
however,
breakout
conventional
filter
daily
does not
a
buy
the
introduced
make
frequent
should
some "fair"
equilibrium.
analysts
or sell
outperform
costs
price
signaled
the
around
market,
new
investors
impending
the
a
Technical
occurs,
can
to
is
barriers.
two barriers
is
eliminate
studying
only
prices
This
past
universally
are
indeed
24
In
the
last
few decades,
have recognized
to
that
be stochastic
Batitsky,
many processes
are
actually
and by Domotor).
can produce
stochastic-like
time-series
of
actually
theorem
observation
of
a
the
topological
system's
chaos
random
does
noise
distinguish
behavior
data
of
initially
eventually
represents
~i+l
continue
to
the
for
the
but
are
random system,
one.
Since
space
while
the
means
When the
dimension
dimension
In contrast,
attractor1s
embedding
to
disordered,
random one.
the
the
space
apparently
space
from
phase
exact
attractor's
value.
the
some
redesigning
space
provides
but
embedding
as
random
phase
and a truly
increase
that
finite-dimension
an asymptotic
&
systems
suggests
redesigned
algorithm
system,
the
a truly
to
a
this
system
as
attaining
data
in
The
in
by Domotor
nonlinear
a procedure
equivalent
a chaotic
increase
thought
by random noise.
the deterministic,
a chaotic
represents
provides
occurs
between
were previously
may appear
signal.
not,
sciences
This
trajectories
single
deterministic
the natural
(see papers
contaminated
the
is
chaotic
processes
(embedding)
portrait
that
behavior.
possibly
The embedding
in
Even low-dimension
economic
chaotic,
scientists
space
will
increases,
when the
dimension
dimension
increases.
We mentioned
the
rate
of
earlier
divergence
positive
Lyapunov
chaos.
A
problem
that
of
exponent
in
the
is
the
Lyapunov
trajectories.
often
estimating
used
Lyapunov
exponent
indicates
The existence
as
an
indication
exponents
is
of
a
of
that
commonly used algorithms
Since
few
Lyapunov
economic
require
series
exponent
a large
of
estimates
such
of
number of
large
size
economic
observations.
are
data
available,
may not
be
so
reliable.
Researchers
economic
data.
demonstrates
since
have applied
A study
that
1960 for
monotonically
than
the
for
they
conclude
their
data.
their
Frank
Lyapunov
since
exchange
than
12)
observed
value
rates
their
as the
the
between
the
authors
forward
rates,
chaos in
have
macroeconomics
computed
series,
chaos.
They
embedding
exponent
the
to
of
of
correlation
dimension
the
(less
series.
They
an asymptotic
Lyapunovexponent
chaotic
predict
these
space
(1992)
and forward
the
achieves
largest
evidence
spot
computed
dimension
but
Hence,
and Sosvilla-Rivero
Lyapunov
less
Lyapunov
cases.
too,
Peseta-Dollar
the
sought
largest
most
(1988),
periods.
of
(1988),
GDP data
of deterministic
Canadian
Spanish
correlation
constitutes
in
of deterministic
2 and 3 and that
This
three-month
various
as well
model,
the
a function
that
Stengos
for
to
space dimension
the
negative
Fernandez-Rodriguez,
at
as
positive.
and
Stengos
of quarterly
that
no evidence
exponents
examined
dimensions
is
analysis
UK, and West Germany increases
found
were
no evidence
Bajo-Rubio,
the
system
and
dimension
also
there
Gencay,
of the embedding
data
that
and have found
have
Japan,
have
of dynamic
Frank,
correlation
Italy,
They
negative
by
as a function
15.
exponents
tools
also
yielded
behavior.
of
root
the
is
Using
one-
apq
mean square
26
errors
walk
lower
than
model.
economic
is
Motivated
the
the
to
the
spectrum
may
periodic
behavior.
although
the
underlying
model
in
the
system,
and
is
data.
to
infinite-dimensional
broad-band
a
frequent
major
whether
of
in
new
way
chaos in
stock
which
is
The power
periodic
or
quasi-
Snyder
(1997)
find
that,
useful
in
forecasting
for
modes in
degrees
of
are
is
the
short-term
systems,
spectra
This
2,
of
the
freedom
of
generated
not
true,
chaotic
power
the
by
an
however,
systems
in
may well
power spectra.
to
UK, World
chaos in
of
some low-dimension
tried
the
of
autocorrelation.
linear
Our group
the
not
Figure
In
1999).
included
a random-
construction
may be advantageous
of
deterministic
and
power
methods
a
the
it
returns
hypothesis
the
generalized
to determine
once we utilized
is
discovery
system.
systems;
systems,
one seen in
generally
broad-band
nine
from
evidence
linear
of
the
trend,
correspond
In order
that
time-series
Saligari
spectrum
exhibit
of
such as the
assist
of
nonlinear
study
computation
long-run
predictions
forecasts
therefore
when analyzing
spectrum,
equivalent
from
inconclusive.
by
point
power
obtained
One may conclude
data
starting
those
stock
stock
indices
excluding
the economy is
dynamic
systems
indices
(Shachmurove,
test
weak-form
by
price
the
to
attempting
data.
of Canada,
USA, France,
chaotic
analyze
Yuen,
to
and Bau,
low-order
analyzed
Europe
Japan,
daily
efficiency
find
The data
Germany,
the
market
Europe-14,
or not,
by us
excluding
the
UK, and
21
c
the
USA,
during
the
1997.
For
the
daily
changes
purposes
in
percentages.
to
U.
The power
spectra
depicted
from
of
our
the
We
converted
were
period
January
1,
report,
we analyzed
stock
price
calculated
S.
Dollars
of
the
1982,
daily
as a function
of
from
Yuen,
of
frequency
(see
1999).
price
Figure
in
returns
and Bau,
stock
5,
relative
expressed
percentages
returns
September
the
indices
(Shachmurove,
the
to
indices
9).
INSERT FIGURE 9
Note
that
spectrum,
this
behavior,
it
is
obtained
the
of
also
returns
lack
power
of
10 depicts
the
is
the
axis
represents
in
consistent
the
with
systems;
power
for
data.
random
example,
1988).
probability
of
The horizontal
axis
periodicity
spectrum
map (Gershefeld,
vertical
have a broad-band
common to many chaotic
from histograms
indices.
daily
implies
type
logistic
Figure
the
of
which
Although
in the
all
distribution
daily
returns
represents
the
of
the
probability
function
the
daily
of
stock
return,
obtaining
as
price
and
this
return.
Insert
Figure
The figure
price
indices
distribution.
10
that
indicates
are
nearly
Although
all
the
Gaussian,
the Gaussian
daily
having
probability
returns
a
of
stock
bell-shaped
distribution
is
28
common in many stochastic
deterministic
chaotiqi
Fi~re
daily
of
of
n.
each
stock
the
with
cell,
finding
is
also
informat$9U
price
exchanges
cubes
in
of
(Pi)
the
Briefly,
n-dimensional
points
the
different
dimension,
it
by some
eXhib£t~d
systems.
11 depicts
returns
dollars,
processes,
indices,
embedding
1;;9; 9p:Hpin
$fi {;~ll
space
E.
of
is
the
estimate
(i).
The
of
of
in
into
the
the
the
US
embedding
divided
One counts
an
Dr,
expressed
as a function
edge size
a point
dimension,
N(E)
number of
probability
information
dimension
I
is
defined
as:
1~
&~
For
a more
Shannon
detailed
(1948).
construction
using
the
largest
susceptible
to
bar
represents
Dr.
Due to
always
Figure
k-sets
noise
the
the
11.
Pi
N(&)
~ i=O
exposition
of
The largest
of
(L
11-k
is
))
log(f:)
the
information
value
sets
generally
of
the
are
considered
(rms)
rms,
of
the
entropy
see
used
the
dimension
the
in
obtained
to
more reliable.
mean square
smallness
Pi
The information
and therefore
root
1og (
be
The vertical
oscillations
vertical
bars
in
are
visible.
INSERT FIGURE 11
Note
oscillations
that,
as
in Dr.
(n)
increases
For example,
so do
(Dr)
and
as n increases
less
the
rms
of
from 1 to
not
the
rms of
the
oscillations
increases
index
and
from
g),
Dr
The
daily
value
of
analysis,
appear
however,
either
German
to
price
(Figures
lla,
the
n,
returns
information
~andom or
a
an
curve
nearly
USA
(Figure
all
of
result
and
asymptote
are
dimension
Figure
of
llf)
dimensions
further.
daily
(Figure
approach
embedding
the
increases
the
the
Once larger
USA indices
that
of
returns
n.
stock
line.
dimensions
11i)
Canadian
In many cases
equals
approximately
a 45-degree
follows
fJq 0.49.
0.01
the
DIc for
in
the
of
at
included
the
large
in
price
indicates
indices
a high-dimension,
the
German and
11 as a whole,
stocks'
a
are
deterministic
process.
Thus,
governed
to
the
by
the
results
a low-dimension,
limited
considered
number
estimate
data
would
weak-form
states
predicted
and thus
by
oil
from
(2000)
futures.
data
such
have
the daily
large
points,
the
that
hypothesis.
prices
and/or
cannot
of
past
price
available
at
similar
earn
consistent
securities
and market
abnormal
information.
conclusions
be
a much larger
with
The weak-form
future
reliably
publicly
arrived
is
should
due
to have a more
dimensions
finding
are not
However,
results
In order
embedding
This
returns
system.
and tentative.
efficiency
current
investors
examining
Ninni
of
be required.
market
hypothesis
for
that
deterministic
as indicative
reliable
set
indicate
the
market
cannot
be
information,
returns
merely
Panas
and
when analyzing
30
Like
physical
aggregate
of
actions
In
allow
needs
preferences
lead
to
structures,
to physical
systems,
desires
influenced
accounts
(see
papers
that
peoples'
random,
this
choices
over
systems
Whether
added
by Krippendorff
change over
reflect
time.
These
such as new inventions
value
regulations.
the
Elementary
do not
do change
changing
for
possibly
that
by many factors
new options,
represent
generalizations.
of physics
that
question
individual,
statistical
and government
model
of
systems
obey laws
and
are
economic
number
for
contrast
perceived
a
vast
however,
time.
that
a
that
particles,
systems,
and political
one can construct
complexity
and also
is
an
open
by Reiner,
Teune
was believed
that
and Tomazinis).
Summary
Before
the
complicated,
complicated
mathematical
(i.e.,
equations
such
Perhaps
is
which
can
differential
complicated,
chaos
models
as the
the
the
or
of
that
by
difference
stochastic-like
be generated
numbers
and/or
or
just
a
equations,
behavior.
of
simple
few
only
by
degrees
of
differential
a large
by stochastic
consequences
relatively
of
partial
equations
equations)
described
large
consisting
most exciting
it
could
with
Navier-Stokes
realization
be
theory,
behavior
models
differential
one of
theory
of
stochastic-like
freedom
of ordinary
advent
dynamic
dynamic
non-linear,
can
exhibit
number
systems.
system
systems,
ordinary
very
!~.
This
observation
dynamic
gives
behavior
can
mathematical
models
equations).
The
complicated
they
models
enable
mechanisms,
control
not
these
One of
to
initial
is
real
and small
systems,
Hence long
chaotic
the
system
is
us
the
systems
term
prediction
is
in
strategies
of
impossible.
to
property
is
a
of
resources
will
not
enable
predictions.
In the
context
of
meteorology,
weather
patterns
is
chaotic
may well
are
the
detailed
lack
is
unknowable
to
long-term
systems
of
when one
of
large
long-term
example,
be),
just
quantum
that
one to generate
for
behavior
of
chaotic
it
not
to perturbations
availability
(as many believe
remain
When one
cornerstone
computational
system
sensitivity
(noise).
The
the
weather
their
are subject
systems,
all.
physical
conditions
chaotic
with
the
is
initial
real
principle
One of
systems
implications
after
Low
variables
devising
practical
deals
of
inexpensively;
important
in
modeling
underlying
perturbations
a fundamental
uncertainty
mechanics.
guide
of chaotic
known and all
predictability
as
the
identifying
of
applications.
processes
into
number
processes.
the hallmarks
and noise.
low-dimensional
small
practical
insights
they
complicated
low-dimensional
simulate
us in
least,
modeling
a
gain
conditions
precisely
of
to
of
to
by
relatively
has tremendous
and assist
processes;
a
us
in some cases,
formulated
possibility
allow
us
be
(i.e.,
behavior
dimension
hope that,
if
the
long-term
and indeterminable,
32
.'
References
Bajo-Rubio,
0.,
"Chaotic
Results
for
39,
207-211.
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Conrad,
J.
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:t:'ambridge,
Fernandez-Rodriguez,
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the
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ct..!
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(1988),
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Atmospheric
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Lorenz,
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~
l",\ci;:.,
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abundance
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"'
of
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a
Ch aotlc
-!q"'y,
0
for
Data
.
a species
£
.
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( 199..1"" )
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"Deterministic
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20, pp. 130-141.
Snooping,
34
LIST
1.c
{\
~!O
OF CAPTIONS
Schematic
description
The flow
rate
time.
(X)
The
different
of the Lorenz
in
figure
initial
3. The chaotic
the
loop
depicts
loop.
is
depicted
two
as a function
time-series
with
of
slightly
conditions.
attractor
constructed
in
three-dimensional
phase
space.
4. Poincare
Figure
description
field
6. The flow
is
difference
,
t=O,
lOT,
9. The
power
when the
and large
20T,
(e):
.e
OJ.;o
stock
the
is
of
and Care
the page.
when a potential
A-C.
images)
period
out
cavity
electrodes
a material
of
the
daily
functions
(c):
Europe
France;
(f):
price
in
directed
B,
obtained
by following
(left),
intermediate
small
blob
at
various
times
t=kT.
30T, 40T, and SOT.
as
The probability
is
A,
stirrer.
(right).
of
spectra
depicted
Europe-14i
1'0;,
c,...1
In
depicted
solution.
(stroboscopic
The deformation
USA
.,
I-
attractor
magneto-hydrodynamic
field
imposed across
tracers
(middle)
the
(streamlines)
sections
Poincare
are
chaotic
electrolyte
The magnetic
electrodes.
passive
of
C contains
The cavity
o.
the
A,
Y
3.
5 . A schematic
ri
i'
section
Germany;
are
of
stock
frequency.
Excluding
distribution
indices
of
returns
the
(g):
UK;
Japan;
functions
depicted
(a):
as
(dJ
:
h):
price
Canada;
World
(b):
Excluding
UK; (i):
of the daily
functions
indices
USA.
returns
the
daily
c~c
~;.
return.
the
UK;
(g):
11.
(a):
price
(a
(d):
indices,
to
i)
the
D1.
Japan;
(i):
root
Canada;
(h):
(c):
USA;
(e):
France;
D1, of
the
dimension,
in
n.
returns
depicted
as
a
the
USA;
11-k
Excluding
Germany;
(e):
(rms)
(c):
exchange
the
sets
were
used
the
Europe
France;
stock
of
The
of
stock
of
function
value
figures.
Europe-14;
USA.
daily
each
mean squared
Excluding
UK; (i):
(f):
for
The largest
of
(b):
Europe
US dollars
is
construction
World
14;
USA.
expressed
the
Europe
Excluding
UK;
dimension,
(a):
(d):
(b):
respectively,
represents
UK;
(h):
information
embedding
in
World
Japan;
The
Canada;
(f):
vertical
oscillations
bar
in
Excluding
the
Germany;
(g):
36
Fig.
1:
Schematic
description
10.
of
the
experimental
A
apparatus.
,
~
§.
~
~
~ "
,~
5.
.
~.
2: The flow
rate
(X) in the
loop
is
depicted
as a function
"F1g.
of time.
The figure
slightly
depicts
different
two time
initial
series
conditions.
resulting
from
~Y.
I
z
Fig.
3:
The chaotic
attractor
constructed
phase space.
in three-dimensional
38
20.0
12.0
8.0
4.0
y
0.0
-4.0
-8.0
X
Fig.
4:
Poincare
sect~~n
of
the
Fig.
chaotic
3.
attractor
depicted
in
39
c
5:
Fig.
A schematic
The cavity
stirrer.
C are
C contains
the
field
magneto
solution.
electrolyte
The magnetic
electrodes.
of
description
is
directed
hydrodynamic
A, B, and
out
of
page.
Fig.
6: The flow
potential
in the cavity
field
difference
is
imposed
a~~,~ss electroqes
when a
A-C.
the
Fig.
following
Poincare
sections
passive
tracers
(stroh,oscopic
when the period
images)
is
obtained
small
by
(left),
.#j'/;::;::;~-::-
({;j'
i
F$g. ,
t=O, lOT,
deformation
20T, 30T, 40T,
of
material
SOT.
blob
at various
times t=kT.
4.1
101;
E 100
tQ)
g. 10-2
'Q)
~ 10-4
a.
10~ U
0.2
c'C
0.4
Frequency (day-I)
102
~ 100
U
~ 100
U
g.
10.2
'-
~ 10.2
~ 10-4
Q.
:4 ~
a.
Q)
Q)
'Q)
Q)
~10-4
0
r
()
0.2
Frequency
.2
10-6
0
02
0.4
Frequency (day.1)
0.4
(day-I)
0.2
0.4
Frequency (day.1)
10Z
10
~ 100
-
~ 100
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Q)
Q)
~ 10.2
g-10
-
E
2 100
u
Q)
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g.
10.2
'-
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~ 10-4
c..
1
0
! 10-4
a.
0
Q)
~
c..
;.:y'
to
10-6
0
0.2
0.4
Frequency(day.1)
The power spectra
Figure
indices
0
are depicted
Europ~
(e) : France;
~."C~
~,
..0
}
,0'
:
(f):
as
of
functions
(g):
c.
Q
Japan;
of
returns
daily
(a):
frequency.
Europe Excluding
Germany;
c;e
0.2
0.4
Frequency (day-1)
UK;
(d):
(h):
UK;
,
World
(i):
stock
Canada;
Excluding
USA.
price
(b) :
USA;
42
Figure
10.
returns
of
return.
(d):
stock
(a):
probability
price
Canada;
Wq~lq Ekcluding
} c
v ,,'
j ';
t
UK;
The
l
C~
USA
"
'*
distribution
indices
(b):
are depicted
Europe
USA; (e):
functions
14;
France;
of
as functions
(c):
Europe
(f):
Germany;
the
of
Excluding
(g):
daily
the
daily
the
Japan;
UK;
(h):
43
(b) Europe14
(a):Canada
10
10
8
8
cS 6
6 6
4
2
4
;
~
()
I
'°0;
Q
$'
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()
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1,0
:c:
Y
O!
8
10
8
USA
6
6
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4
q
4
2
c
1"
n,
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10
in
n
0
c;
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5
"
0
0'
;~1
;\1
~(}
,;Ii
~
n
Ii
n
co"
Figure
stock
11.
'"
The information
price
indices,
depicted
as a function
k
sets
value
vertical
Canada;
Excluding
USA.
bar
were
expressed
of the
used
represents
(b):
USA;
Europe
(e):
dimension,
in
the
14;
France;
Dr,
in US dollars,
embedding
the
of
Europe
(f):
the
dimension,
the
n.
of
the
oscillations
Excluding
Germany;
daily
of different
construction
rms
(c):
of
(g):
the
Japan;
returns
of
exchange
is
The largest
figures.
11The
in
Dr.
(a):
UK;
(d):
World
(h):
UK;
(i):