Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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of
a
two-dimensional
disk
as
a
flow
in
an
orientable
three
dimensional
manifold.
Such
a
suspension
occurs
frequently
in
periodically
forced
nonlinear
oscillators,
such
as
the
Duffing
equation.
From
this
suspension
they
constructed
a
template,
which
is
a
branched
two-manifold
with
a
semiflow,
in
such
a way
that
the
periodic
orbits
are
isotopic
to
those
in
the
full
three-dimensional
flow.
Some
existence,
uniqueness,
and
nonexistence
theorems
for
families
of
torus
knots
are
obtained.
They
then
were
able
to
connect
a
countable
subsequence
of
one-dimensional
bifurcations
with
a
subsequence
of
area-preserving
bifurcations
in
a
two
parameter
family
of
suspensions
by
using
knot
theory,
kneading
theory
and
Hamiltonian
bifurcation
theory.
The
moral
of
the
results
is
that
there
are
no
universal
routes
to
chaos!
For
more
recent
work
on
the
classification
of
knotted
periodic
orbits
in
periodically
forced
nonlinear
oscillators,
as
well
as
a
review
of
earlier
work,
see
Holmes
[1988].
Recently,
Agarwal,
Banerjee
and
Bhattacharjee
[1986]
have
shown
that
at
the
threshold
of
period-doubling
chaos
in
a
dynamical
system
such
as
the
Lorenz
system,
the
fractal
dimension
of
the
associated
strange
attractor
assumes
a
universal
value.
Dekker
[1986]
using
computer
simulation
demonstrated
that
the
tunnelling
orbits
under
a
symmetric
double-well
potential
can
be
assigned
a
fractal
dimension.
The
three-dimensional
Lorenz
flow
is
approximated
by
a
two-dimensional
flow
with
a
branch
curve
with
the
use
of
the
approximation
of
the
Lorenz
attractor
by
invariant
two-
dimensional
manifolds
[Dorfle
and
Graham
1983],
and
the
probability
density
generated
by
the
flow
on
the
invariant
manifolds
in
the
steady
state
is
obtained.
It
is
also
shown
that
the
probability
density
arises
in
the
Lorenz
model
subject
to
stochastic
forces
as
a
self-consistent
approximation
for
very
small
but
finite
noise.
Recently,
Lahiri
and
Nag
[1989]
have
considered
intermittency
in
inverted-pitchfork
bifurcation
of
a
10
dissipative
map
and
of
a
20
conservative
map.
They
have
also
294
considered
the
effect
of
noise
and
the
scaling
laws
in
the
presence
of
noise.
Furthermore,
they
have
obtained
the
results
for
the
saddle-node
bifurcation
in
20
area-
preserving
maps.
For
a
broader
view
of
chaos,
with
its
implications,
see,
for
instance,
Holden
[1986],
and
Schuster
[1984].
For
a
more
current
review
of
chaotic
phenomena
in
nonlinear
systems,
see
Hao
[1988].
For
chaotic
phenomana
in
astrophysics,
see
for
instance
Buchler
and
Eichhorn
[1987],
Contopoulos
[1985].
For
nonlinear
dynamical
phenomena
in
chemical
systems,
see
for
instance,
Vidal
and
Pacault
[1984].
For
fluid
or
plasma
phenomena,
see
for
instance,
Marsden
[1984],
Sagdeev
[1984],
Rand
and
Young
[1981].
For
thoes
readers
who
are
interested
in
computer
graphic
presentations
and
numerical
integration
of
nonlinear
dynamical
systems,
there
are
several
PC
compatible
software
available
for
such
purpose.
Although,
we
have
mainly
concentrated
on
nonlinear
dynamical
systems
represented
by
ordinary
differential
equations,
similar
techniques
can
also
be
applied
to
nonlinear
dynamical
systems
represented
by
partial
differential
equations.
Birnir
and
Morrison
[1987]
have
discussed
structural
stability
and
chaotic
solutions
of
perturbed
Benjamin-One
equations
for
characterizing
the
turbulent
motion.
Bishop
et
al
[1983,
1988]
have
discussed
chaotic
solutions
to
other
perturbed
nonlinear
partial
differential
systems
such
as
Sine-Gordon,
and
nonlinear
Schrodinger
equations.
Olsen
and
Samuelsen
[1987]
have
examined
the
one-dimensional
sine-Gordon
soliton
equation
in
the
presence
of
driving
and
damping
numerically.
They
have
found
that
by
increasing
the
driving
strength,
the
system
exhibits
an
infinite
sequence
of
period
doubling
leading
to
chaos.
They
noticed
that
the
first
bifurcation
occured
after
a
narrow
regime
characterized
by
intermittency-type
of
chaos
and
quasi-periodic
oscillations.
They
further
attributed
the
origin
of
the
intermittency-type
of
chaos
by
the
competition
between
two
spatial
patterns
described
by
the
presence
of
295
one
or
two
breather-like
modes
respectively.
Brandstater
et
al
[1983]
have
presented
evidence
for
low
dimensional
strange
attractors
in
Couette-Taylor
flow
data.
Doering
et
al
[1987]
have
obtained
an
exact
analytic
computation
of
the
Liapunov
dimension
of
the
universal
attractor
of
the
complex
Ginzburg-Landau
equation
for
a
finite
range
of
parameter
values.
Meanwhile,
Sirovich
and
Rodriguez
[1987]
have
used
the
Ginzburg-Landau
equation
to
extract
a
complete
set
of
uncorrelated
coherent
structures,
which
are
then
used
as
a
basis
for
the
dynamical
description
of
coherent
structures
in
the
attractor
set.
Chate
and
Manneville
[1987]
discuss
the
transition
to
turbulence
via
spatiotemporal
intermittency
observed
in
a
partial
differential
equation
displays
statistical
features.
The
transition
to
chaos
through
intermittency
has
recently
been
observed
in
a
steady-state
plasma
[Cheung,
donovan
and
Wong
1988].
Even
the
traditional
problems,
such
as
the
Navier-Stokes
equations
for
an
incompressible
fluid
on
a
two-dimensional
torus,
are
rich
in
phenomenology.
For
instance,
Franceschini
[1983a,b)
has
numerically
investigated
two
truncations
of
the
above
problem.
Over
a
decade
of
active
research,
a
number
of
experiments
have
concluded
that
the
production
of
complex,
unpredictable
turbulence
for
low-dimensional
systems
is
due
to
few-dimensional
attractors.
Indeed,
based
on
these
and
other
observations
and
analysis,
the
"conventional
wisdom"
has
concluded
that
the
Newhouse-Ruelle-Takens
route
of
few-dimensional
chaos
[Ruelle
and
Takens
1971,
Newhouse,
Ruelle
and
Takens
1978]
is
the
proper
explanation
of
the
nature
of
turbulence.
Nonetheless,
there
are
several
alternative
explanations
of
turbulence
which
do
not
employ
the
attractive
hypothesis,
namely,
the
spin-glass
relaxation
[Walden
and
Ahlers
1980,
Crutchfield
1984],
spatial
noise
amplification
[Deissler
and
Kaneko
1987],
and
transients
[Crutchfield
and
Kaneko
1988].
Let
us
briefly
describe
three
most
common
scenarios
for
turbulence.
By
the
way,
here
chaos
and
weak
turbulence
are
interchangeable.
296
The
Newhouse-Ruelle-Takens
(NRT)
scenario
is
the
oldest
one
in
describing
the
route
to
turbulence.
This
can
be
stated
in
the
following
theorem
[Newhouse,
Ruelle
and
Takens
1978):
Theorem
6.7.1
Let
x
be
a
constant
vector
field
on
the
n-dim
torus
T" =
R"/Z".
If
n
~
3,
every
c•
neighborhood
of
x
contains
a
vector
field
x'
with
a
strange
Axiom
A
attractor.
If
n
~
4,
we
may
take
~
for
c•.
Under
all
these
assumptions,
the
NRT
scenario
asserts
that
a
strange
attractor
is
likely
to
occur
in
the
following
sense.
In
the
space
of
all
differential
equations,
some
equations
have
strange
attractors,
and
others
have
none.
Those
which
do
have
strange
attractors
form
a
set
containing
a
subset
which
is
open
in
the
c•
topology.
The
closure
of
this
open
set
contains
the
constant
vector
fields
on
the
torus
T
3
•
The
measurable
consequences
of
the
presence
of
strange
attractor
following
the
NRT
scenario
is
the
following:
If
a
system
starting
from
a
stationary
state
solution
undergoes
three
Hopf
bifurcations
when
a
parameter
is
varied,
then
it
is
very
likely
that
the
system
possesses
a
strange
attractor
with
sensitivity
to
initial
conditions
after
the
third
bifurcation.
And
the
power
spectrum
of
such
a
system
will
exhibit
first
one,
then
two,
and
possibly
three
independent
basic
frequencies.
If
there
is
a
strange
attractor,
when
the
third
frequency
is
about
to
appear
some
broad-band
noise
will
simultaneously
also
appear.
When
these
happens,
we
consider
the
system
chaotic
or
turbulent.
Then
it
is
natural
to
ask
whether
or
not
three-frequency
quasiperiodic
orbits
are
to
be
expected
in
typical
nonlinear
dynamical
systems?
This
is
the
question
raised
by
Grebogi,
Ott
and
Yorke
[1983].
Nonetheless,
at
that
time
the
answer
was
positive
but
incomplete.
Another
scenario
leading
to
turbulence
is
the
Feigenbaum
scenario.
These
type
of
systems
usually
can
be
reduced
to
Poincare
return
maps
or
area-preserving
maps,
and
the
bifurcations
of
the
orbit
structure
are
pitchfork
bifurcations,
i.e.,
a
stable
fixed
point
loses
its
297
stability
and
gives
rise
to
a
stable
periodic
orbit
as
a
parameter
is
changed.
This
corresponds
to
a
crossing
of
one
eigenvalue
of
the
tangent
map
through
-1.
In
an
experiment,
if
one
observes
subharmonic
bifurcations
at
~,,
~
2
,
then
according
to
the
scenario,
it
is
very
probable
for
a
further
bifurcation
to
occur
near
~
3
=
~
2
-
(~
1
-
~
2
)/6,
where
6 =
4.66920
•••
is
universal
[Feigenbaum
1978, 1979,
1980;
Collet
and
Eckmann
1980,
Greene,
MacKay,
Vivaldi
and
Feigenbaum
1981).
Some
other
universal
fine
structures
in
periodic
doubling
systems
have
also
been
discussed
by
Geisel
and
Nierwetberg
[1981,
1982].
Furthermore,
if
one
has
seen
three
bifurcations,
a
fourth
one
becomes
more
probable
then
the
third
one
after
the
first
two,
etc.
And
at
the
accumulation
point,
one
will
observe
aperiodic
behavior,
but
without
broad-band
spectrum.
This
scenario
is
extremely
well
tested
both
numerically
and
experimentally.
The
periodic
doublings
have
been
observed
in
most
low
dimensional
dynamical
systems.
Giglio,
Musazzi
and
Perini
[1981]
presented
experimental
results
of
a
Rayleigh-Benard
cell.
By
appropriate
preparation
of
the
initial
state,
the
system
can
be
brought
into
a
single
frequency
oscillatory
regime.
When
a
further
increase
of
the
temperature
gradient
makes
the
system
undergo
a
reproducible
sequence
of
period-doubling
bifurcations,
the
Feigenbaum
universal
numbers
are
also
determined.
Gonzalez
and
Piro
[1983]
considered
a
nonlinear
oscillator
externally
driven
by
an
impulsive
periodic
force,
and
an
exact
analytical
expression
for
the
Poincare
map
for
all
values
of
parameters
is
obtained.
This
model
also
displays
period-doubling
sequences
of
chaotic
behavior
and
the
convergence
rate
of
these
cascades
is
in
good
agreement
with
the
Feigenbaum
theory.
The
third
scenario
is
the
Pomeau-Manneville
scenario
[Pomeau
and
Manneville
1980],
which
is
also
termed
transition
to
turbulence
through
intermittency.
The
mathematical
status
of
this
scenario
is
less
than
satisfactory.
Nonetheless,
this
scenario
is
associated
with
a
saddle-node
bifurcation,
i.e.,
the
amalgamation
of
a
298
stable
and
an
unstable
fixed
point.
One
can
state
this
scenario
for
general
dynamical
systems
as
follows.
Assume a
one-
parameter
family
of
dynamical
systems
has
Poincare
maps
close
to
a
one-parameter
family
of
maps
of
the
interval,
and
that
these
maps
have
a
stable
and
unstable
fixed
point
which
amalgamate
as
the
parameter
is
varied.
As
the
parameter
is
varied
further
to~
=
1.75
from
the
critical
parameter
value
~c'
one
will
see
intermittently
turbulent
behavior
of
random
duration.
The
difficulty
with
this
scenario
is
that
it
does
not
have
any
clear-
cut
or
visible
precursors
like
the
other
two
scenarios.
Recently,
Keeler
and
Farmer
[1987]
investigated
a
one
dimensional
lattice
of
coupled
quadratic
maps.
They
found
that
the
motion
of
the
spatial
domain
walls
causes
spatially
localized
changes
from
chaotic
to
almost
periodic
behavior
and
the
almost
periodic
phases
have
eigenvalues
very
close
to
one
and
with
a
1/f
low
frequency
spectrum.
This
behavior
has
some
aspects
of
Pomeau-
Manneville
intermittency,
but
is
quite
robust
under
changes
of
parameters.
We
would
like
to
emphasize
that
since
a
given
dynamical
system
may
have
many
attractors,
several
scenarios
may
evolve
concurrently
in
different
regions
of
phase
space.
Therefore,
it
is
natural
if
several
scenarios
occur
in
a
given
system
depending
on
how
the
initial
state
of
the
system
is
prepared.
Furthermore,
the
relevant
parameter
ranges
may
overlap,
thus
although
the
basins
of
attraction
for
different
scenarios
must
be
disjoint,
they
maybe
interlaced.
Recently,
Grebogi,
Kostelich,
ott
and
Yorke
[1986]
using
two
examples
to
show
that
basin
boundary
dimensions
can
be
different
in
different
regions
of
phase
space.
They
can
be
fractal
or
nonfractal
depending
on
the
region.
In
addition,
they
have
shown
that
these
regions
of
different
dimension
can
be
intertwined
on
an
arbitrarily
fine
scale.
They
further
conjectured
that
a
basin
boundary
typically
can
have
at
most
a
finite
number
of
possible
dimensions.
Indeed,
a
series
of
papers
by
Gollub
and
coworkers
using
laser-Doppler
methods
have
identified
four
299
distinct
sequences
of
instabilities
leading
to
turbulence
at
low
Prandtl
number
(between
2.5
to
5.0)
in
fluid
layers
of
small
horizontal
extent
[Gollub
and
Benson
1980,
Gollub
and
McCarriar
1982,
Gollub,
McCarriar
and
Steinman
1982].
McLaughlin
and
Martin
[1975]
proposed
a
mathematical
theme
of
the
transition
to
turbulence
in
statically
stressed
fluid
systems.
Systems
are
classified
according
to
Hopf
bifurcation
theorem.
They
have
found
certain
kind
of
flows
obey
the
Boussinesq
conditions
exhibit
hysteresis,
finite-amplitude
instabilities,
and
immediate
transition
to
turbulence.
They
have
also
found
another
kind
of
flow,
such
as
a
model
of
fluid
convection
with
a
low-Prandtl
number,
in
which
as
the
stress
increases,
a
time-periodic
regime
precedes
turbulence.
Nonetheless,
the
transition
to
nonperiodic
behavior
in
this
model
is
found
to
proceed
in
accordance
with
the
NRT
scenario.
Although
we
have
not
discussed
stochastic
influence
on
the
nonlinear
dynamical
systems,
there
are
many
interesting
results
in
this
area.
For
instance,
Horsthemke
and
Malek-Mansour
[1976],
using
the
method
of
Ito
[1944,
1951]
stochastic
differential
equations,
have
shown
that
in
the
vicinity
of
the
bifurcation
point
the
external
noise,
even
they
are
characterized
by
a
small
variance,
can
influence
profoundly
the
macroscopic
behavior
of
the
system
and
gives
rise
to
new
phenomena
not
predicted
by
the
deterministic
analysis.
In
Ruelle
[1985],
the
ergodic
theory
of
differentiable
dynamical
systems
is
reviewed,
and
it
is
applied
to
the
Navier-Stokes
equation.
He
also
obtains
upper
bounds
on
characteristic
exponents,
entropy,
and
Haussdorff
dimension
of
attracting
sets.
Recently,
Machacek
[1986]
presented
a
general
method
for
calculating
the
moments
of
invariant
measure
of
multidimensional
dissipative
dynamical
systems
with
noise.
In
particular,
moments
of
the
Lorenz
model
are
calculated.
Nicolis
and
Nicolis
[1986]
have
casted
the
Lorenz
equations
in
the
form
of
a
single
stochastic
differential
equation
where
a
deterministic
part
representing
a
bistable
dynamical
system
is
forced
by
a
300
noise
process.
An
analytically
derived
fluctuation-
dissipation-like
relationship
linking
the
variance
of
the
noise
to
the
system's
parameters
provides
a
satisfactory
explanation
of
the
numerical
results.
This
leads
the
authors
to
suggest
that
the
emergence
of
chaos
in
the
Lorenz
model
is
associated
with
the
breakdown
of
the
time
scale
separation
between
different
variables.
This
gives
some
credence
to
the
frequent
assertion
in
the
literature
that
the
elimination
of
fast
variables
generates
noise
in
the
subset
of
the
slow
variables.
Horsthemke
and
Lefever
[1977]
have
demonstrated
on
a
chemical
dynamics
model
that
even
though
the
system
is
above
the
critical
point,
phase
transitions
can
still
be
induced
solely
by
the
effect
of
external
noise.
Jeffries
and
Wiesenfeld
[1985)
have
measured
the
power
spectra
of
a
periodically
driven
p-n
junction
in
the
vicinity
of
a
dynamical
instability.
They
have
found
that
the
addition
of
external
noise
introduces
new
lines
in
the
spectra,
which
become
more
prominent
as
a
bifurcation
(either
period
doubling
or
Hopf
bifurcation)
is
approached.
Furthermore,
they
have
found
that
the
scaling
of
the
peak,
width,
area
and
lineshape
of
these
lines
are
in
excellent
agreement
with
the
predictions
[Wiesenfeld
1985).
Experimentally,
the
onset
of
chaos
in
dynamical
systems
can
often
be
analyzed
in
terms
of
a
discrete-time
map
with
a
quadratic
extremum
[Collet
and
Eckmann
1980].
Kapral
and
Mandel
[1985)
investigated
the
bifurcation
structure
of
a
nonautonomous
quadratic
map.
They
found
that
alghough
nontrivial
fixed
points
do
not
exist
for
such
system,
a
bifurcation
diagram
can
be
constructed
provided
that
the
sweep
rate
is
not
too
large.
Morris
and
Moss
[1986]
constructed
an
electronic
circuit
model
of
a
nonautonomous
quadratic
map,
different
from
the
one
by
Kapral
and
Mandel
[1985).
They
observed
that
bifurcation
points
are
postponed
by
amounts
which
depend
on
both
the
sweep
velocity
and
the
order
of
the
bifurcation.
The
measured
results
obey
two
scaling
laws
predicted
by
Kapral
and
Mandel
[1985)
and
provide
evidence
for
the
universality
301
of
these
scaling
laws.
Newell,
Rand
and
Russell
(1988)
recently
suggested
that
the
transport
properties
and
dissipation
rates
of
a
wide
class
of
turbulent
flows
are
determined
by
the
random
occurrence
of
coherent
events
that
correspond
to
certain
orbits
in
the
non-
compact
phase
space
which
are
attracted
to
special
orbits
which
connect
saddle
points
in
the
finite
region
of
phase
space
to
infinity
and
represent
coherent
structures
in
the
flow
field.
Recently,
Chernikov
et
al
(1987]
have
proposed
that
chaotic
web may
be
able
to
explain
the
origin
of
ultra
high
energy
cosmic
rays.
Eckmann
and
Ruelle
(1985)
gave
an
interesting
review
of
the
mathematical
ideas
and
their
concrete
implementation
in
analyzing
experiments
in
ergodic
theory
of
dynamical
systems.
Recently,
Ruelle
(1987]
suggested
the
study
of
time
evolutions
with
adiabatically
fluctuating
parameters,
i.e.,
evolutions
of
the
form
xt+l
=
f(x,~(t))
for
discrete
time.
Similar
to
the
nonlinearly
coupled
oscillators,
Yuan,
Tung,
Feng
and
Narducci
(1983)
analyzed
some
general
features
of
coupled
logistic
equations.
They
have
found
that
for
selected
values
of
the
three
control
parameters,
chaotic
behavior
may
emerge
out
of
an
infinite
sequence
of
period-doubling
bifurcations.
There
also
exist
large
regions
of
control
parameter
space,
where
two
characteristic
frequencies
exist,
and
the
approach
to
chaos
follows
the
NRT
scenario.
Like
the
coupled
oscillators,
as
the
ratio
of
these
frequencies
is
varied,
phase
locking
and
quasiperiodic
orbits
are
observed
enroute
to
chaos.
Recently,
Klosek-Dygas,
Matkowsky
and
Schuss
(1988]
have
considered
the
stochastic
stability
of
a
nonlinear
oscillator
parametrically
excited
by
a
stationary
Markov
process.
Harikrishnan
and
Nandakumaran
(1988)
recently
analyzed
numerically
the
bifurcation
structure
of
a
two-dimensional
noninvertible
map
of
the
form
suggested
by
Ruelle
and
they
showed
that
different
periodic
cycles
are
arranged
in
it
in
the
same
order
as
in
the
logistic
map.
Indeed,
this
map
302
satisfies
the
general
criteria
for
the
existence
of
Sarkovskii
[1964]
ordering,
which
is
simply
the
order
in
which
different
period
cycles
are
arranged
along
the
parameter
axis.
On
the
other
hand,
the
Henan
[1976]
map
can
be
considered
as
the
two-dimensional
analogue
of
the
quadratic
map,
yet
the
Sarkovskii
theorem
does
not
hold
[Devaney
1986].
One
may
also
expect
that
external
noise
can
influence
the
nature
of
chaotic
systems.
Surprisingly,
the
theorem
due
to
Kifer
[1974]
states
that
for
a
dynamical
system
with
an
Axiom
A
attractor,
the
system
is
insensitive
to
small
external
noise.
This
is
experimentally
demonstrated
for
Rayleigh-Benard
system
by
Gollub
and
Steinman
[1980].
Ruelle
[1986a,b)
expanded
to
consider
the
analytic
properties
of
the
power
spectrum
for
Axiom
A
systems
near
resonance.
Recently,
Ciliberto
and
Rubio
[1987]
studied
experimentally
the
spatial
patterns
in
temporal
chaotic
regimes
of
Rayleigh-Benard
convection.
Recently,
cumming
and
Linsay
[1987],
using
a
simple
operational-amplifier
relaxation
oscillator
driven
by
a
sine
wave
which
can
be
varied
in
frequency
and
amplitude,
have
presented
experimental
evidence
for
deviations
from
universality
in
the
transition
to
chaos
from
quasiperiodicity
in
a
nonlinear
dynamical
system.
In
fact,
they
have
shown
that
the
power
spectrum,
tongue
convergence
rate,
and
spectrum
of
critical
exponents
all
differ
from
the
theory.
Meanwhile,
in
a
different
domain,
cvitanovic
[1985]
argued
that
period
doubling
and
mode
lockings
for
circle
maps
are
characterized
by
universal
scaling
and
can
be
measured
in
a
variety
of
nonlinear
systems.
Indeed,
such
phenomena
not
only
have
been
observed
in
physical
phenomena,
but
also
have
been
observed
in
biological
and
physiological
situations.
For
instance,
the
spontaneous
rhythmic
activity
of
aggregates
of
embryonic
chick
heart
cells
was
perturbed
by
the
injection
of
single
current
pulses
and
periodic
trains
of
current
pulses.
The
regular
and
irregular
dynamical
behavior
produced
by
periodic
stimulation,
as
303