Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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period
doubling
bifurcations,
were
predicted
theoretically
and
observed
experimentally
by
Guevara,
Glass
and
Shrier
[1981].
The
above
paragraphs
show
that
even
though
we
have
some
understanding
of
low
dimensional
(one-
or
two-dimensional)
chaos,
but
there
is
still
much
more
we
do
not
understand
and
also
with
lots
of
confusion,
to
say
the
least!
In
the
past
several
years,
a
great
deal
of
interest
has
been
focused
on
the
temporal
evolution
and
behavior
of
chaos,
which
arises
in
spatially
constrained
macroscopic
systems
while
some
control
parameters
are
varied.
Recently,
it
has
become
increasingly
more
popular
to
study
the
spatial
pattern
formation
and
the
transition
to
turbulence
in
extended
systems
[Thyagaraja
1979,
Wesfreid
and
Zaleski
1984,
Bishop,
Campbell
and
Channell
1984],
and
bifurcations
in
systems
with
symmetries
[Stewart
1988
and
references
cited
therein,
Sattinger
1983],
and
nonlinear
mode
competition
between
instability
waves
in
the
forced
free
shear
layer
[Treiber
and
Kitney
1988].
Recently,
Pismen
[1987]
has
investigated
the
strong
influence
of
periodic
spatial
forces
on
the
selection
of
stationary
patterns
near
a
symmetry-breaking
bifurcation.
Using
simple
model
equations
of
long-scale
thermal
convection,
he
demonstrated:
(i)
the
transition
between
alternative
patterns,
(ii)
emergence
of
spatially
quasiperiodic
patterns,
and
(iii)
evolution
of
patterns
due
to
rotating
phases.
The
presence
of
symmetry
usually
can
simplify
the
analysis,
yet
symmetry
can
also
lead
to
more
complicated
dynamical
behavior.
Swift
and
Wiesenfeld
[1984]
have
examined
the
role
of
symmetry
in
systems
displaying
period-doubling
bifurcations.
They
have
found
that
symmetric
orbits
usually
will
not
undergo
period-doubling,
and
those
exceptional
cases
cannot
occur
in
a
large
class
of
systems,
including
the
sinusoidally
driven
damped
oscillators,
and
the
Lorenz
model.
Experimentally,
conservation
laws
are
manifested
through
the
observation
of
restrictions
on
transport
of
chaotic
orbits
in
phase
space.
Recently,
Skiff
et
al
[1988]
have
presented
experiments
304
which
demonstrated
the
conservation
of
certain
integrals
of
motion
during
Hamiltonian
chaos.
Recently,
Coullet,
Elphick
and
Repaux
[1987]
presented
two
basic
mechanisms
leading
to
spatial
complexity
in
one-dimensional
patterns,
which
are
related
to
Melnikov's
theory
of
periodically
driven
one-
degree-of-freedom
Hamiltonian
systems,
and
Shilnikov's
theory
of
two-degrees-of-freedom
conservative
systems
[see,
for
instance,
Guckenheimer
and
Holmes
1984].
It
is
interesting
to
note
that
Coullet
and
Elphick
[1987]
have
used
Melnikov•s
analysis
to
construct
recurrence
time
maps
near
homoclinic
and
heteroclinic
bifurcations,
and
they
have
shown
that
an
elegent
method
developed
by
Kawasaki
and
Ohta
[1983]
to
study
defect
dynamics
is
equivalent
to
Melnikov's
theory.
Recently,
Newton
[1988]
considered
the
spatially
chaotic
behavior
of
separable
solutions
of
the
perturbed
cubic
Schrodinger
equation
in
certain
limits.
Unfortunately,
the
limits
under
consideration
were
not
applicable
to
nonlinear
optics,
otherwise,
its
applications
would
be
much
broader.
On
the
subject
of
Hamiltonian
systems,
recently
Moser
[1986]
gave
a
very
interesting
and
broad
review
of
recent
developments
in
the
theory
of
Hamiltonian
systems.
Although
many
real
world
problems
are
dissipative,
nonetheless,
as
we
have
pointed
out
in
Ch.
2,
Hamiltonian
systems
can
give
us
some
global
information
on
the
structure
of
the
solutions.
Hopefully,
we
can
treat
the
dissipative
or
dispersive
systems
as
perturbations
of
Hamiltonian
systems
and
use
KAM
theory
to
determine
the
characteristics
of
the
solutions.
On
the
subject
of
KAM
theory,
it
is
known
that
the
KAM
tori
survive
small
but
finite
perturbation
and
are
expected
to
break
at
a
critical
value
of
the
parameter
which
depends
upon
the
frequency.
Recently,
Farmer
and
Satija
[1985]
and
Umberger,
Farmer
and
Satija
[1986]
studied
the
breakup
of
a
two-dimensional
torus
described
by
a
critical
circle
map
for
arbitrary
winding
numbers.
They
have
demonstrated
that
such
a
breakdown
can
be
described
by
a
chaotic
renormalization
group
where
the
chaotic
renormalization
orbits
converge
on
a
305
strange
attractor
of
low
dimension.
And
they
have
found
that
the
universal
exponents
characterizing
this
transition
are
global
quantities
which
quantify
the
strange
attractor.
Recently,
MacKay
and
van
Zeijts
[1988]
have
found
universal
scaling
behavior
for
the
period-
doubling
tree
in
two-parameter
families
of
bimodal
maps
of
the
interval.
A
renormalization
group
explanation
is
given
in
terms
of
a
horseshoe
with
a
Cantor
set
of
two-dimensional
unstable
manifolds
instead
of
the
usual
fixed
point
with
one
unstable
direction.
Satija
[1987]
recently
has
found
that
similar
results
are
also
valid
in
Hamiltonian
systems,
i.e.,
the
breakup
of
an
arbitrary
KAM
orbit
can
be
described
by
a
universal
strange
attractor,
and
the
global
critical
exponents
can
characterize
the
breakup
of
almost
all
KAM
tori.
Numerically,
Sanz-Serna
and
Vadillo
[1987]
considered
the
leap-frog
(explicit
mid-point)
discretization
of
Hamiltonian
systems,
and
it
is
proved
that
the
discrete
evolution
preserves
the
symplectic
structure
of
the
phase
space.
Under
suitable
restrictions
of
the
time
step,
the
technique
is
applied
to
KAM
theory
to
guarantee
the
boundedness
of
the
computed
points.
Dimension
is
one
of
the
basic
properties
of
an
attractor.
Farmer,
Ott
and
Yorke
[1983]
discussed
and
reviewed
different
definitions
of
dimension,
and
computed
their
values
for
typical
examples.
They
have
found
that
there
are
two
general
types
of
definitions
of
dimension,
those
that
depend
only
on
metric
properties
and
those
that
depend
on
the
frequency
with
which
a
typical
orbit
visits
different
regions
of
the
attractor.
They
have
also
found
that
the
dimension
depending
on
frequency
is
typically
equal
to
the
Liapunov
dimension,
which
is
defined
by
Liapunov
numbers
and
easier
to
calculate.
An
algorithm
which
calculates
attractor
dimensions
by
the
correlation
integral
of
the
experimental
time
series
was
proposed
by
Grassberger
and
Procaccia
[1983].
Nonetheless,
due
to
nonstationarity
of
the
system,
consistent
results
may
require
using
a
small
306
data
set
and
covering
short
time
intervals
during
which
the
system
can
be
approximately
considered
stationary.
Under
such
circumstances,
several
sources
of
error
in
calculating
dimensions
may
result.
Recently,
Smith
[1988]
has
computed
intrinsic
limits
on
the
minimum
size
of
a
data
set
required
for
calculation
of
dimensions.
A
lower
bound
on
the
number
of
points
required
for
a
reliable
estimation
of
the
correlation
exponent
is
given
in
terms
of
the
dimension
of
the
object
and
the
desired
accuracy.
A
method
of
estimating
the
correlation
integral
computed
from
a
finite
sample
of
a
white
noise
is
also
obtained.
Havstad
and
Ehlers
[1989]
have
proposed
a
method
to
resolve
such
problems.
Eckmann,
Ruelle
and
Ciliberto
[1986]
discussed
in
detail
an
algorithm
for
computing
Liapunov
exponents
from
an
experimental
time
series,
and
a
hydrodynamic
experiment
is
investigated
as
an
example.
Recently,
Ellner
[1988]
has
come
up
with
a
maximum-likelihood
method
for
estimating
an
attractor•s
generalized
dimensions
from
time-series
data.
Meanwhile,
Braun
et
al
[1987],
Ohe
and
Tanaka
[1988]
have
investigated
the
ionization
instability
of
weakly
ionized
positive
columns
of
helium
glow
discharges,
and
the
transition
to
the
turbulent
state
is
experimentally
observed.
By
calculating
the
correlation
integral,
they
have
determined
that
the
periodic
instability
has
low
dimensionality
while
turbulent
instability
has
a
higher,
but
finite,
dimensionality.
Nonetheless,
they
have
not
determined
the
dimensions
of
strange
attractors
in
their
system
by
computing
the
Liapunov
exponents
for
the
experimental
time
series.
This
can
be
an
interesting
problem
to
relate
the
dimensions
obtained
via
two
approaches.
Earlier,
Farmer
[1982]
studied
the
chaotic
attractors
of
a
delay
differential
equation
and
he
has
found
that
the
dimension
of
several
attractors
computed
directly
from
the
definition
of
dimensions
agrees
to
within
experimental
resolution
of
the
dimension
computed
from
the
spectrum
of
Liapunov
exponents
according
to
a
conjecture
of
Kaplan
and
Yorke
[1979].
Farmer
and
Sidorowich
[1987]
have
presented
a
technique
which
allows
us
to
make
short-term
307
predictions
of
the
future
behavior
of
a
chaotic
time
series
using
information
from
the
past.
Recently,
Tsang
[1986]
proposed
an
analytical
method
to
determine
the
dimensionality
of
strange
attractors
in
two-dimensional
dissipative
maps.
In
this
method,
the
geometric
structures
of
an
attractor
are
obtained
from
a
procedure
developed
previously.
From
the
geometric
structures,
the
Hausdorff
dimension
for
the
Cantor
set
is
determined
first,
then
for
the
attractor.
The
results
have
compared
well
with
numerical
results.
Recently
Auerbach
et
al
[1987]
were
able
to
approximate
the
fractal
invariant
measure
of
chaotic
strange
attractors
by
the
set
of
unstable
n-periodic
orbits
of
increasing
n,
and
they
also
presented
algorithms
for
extracting
the
periodic
orbits
from
a
chaotic
time
series
and
for
calculating
their
stabilities.
The
topological
entropy
and
the
Hausdorff
dimension
can
also
be
calculated.
For
nonlinear
and
nonstationary
time
series
analysis,
see
Priestley
[1988].
On
the
other
hand,
it
is
important
to
realize
that
there
are
other
types
of
invariant
sets
in
dynamical
systems
that
are
not
attracting,
and
these
non-attracting
invariant
sets
also
play
a
fundamental
role
in
the
understanding
of
dynamics.
For
instance,
these
non-attracting
invariant
sets
occur
as
chaotic
transient
sets
[Grebogi,
ott
and
Yorke
1983,
Kantz
and
Grassberger
1985,
Szepfalusy
and
Tel
1986],
fractal
basin
boundaries
[McDonald,
Grebogi,
Ott
and
Yorke
1985],
and
adiabatic
invariants
[Brown,
Ott
and
Grebogi
1987].
In
many
cases,
these
invariant
sets
have
complicated,
Cantor
set-like
geometric
structures
and
almost
all
the
points
in
the
invariant
sets
are
saddle
points.
Hsu,
ott
and
Yorke
[1988]
called
such
non-attracting
sets
strange
saddles.
They
discussed
and
numerically
tested
formula
relating
the
dimensions
of
strange
saddles
and
their
stable
and
unstable
manifolds
to
the
Liapunov
exponents
and
found
to
be
consistent
with
the
conjecture
formulated
by
Kaplan
and
Yorke
[1979].
Recently,
Lloyd
and
Lynch
[1988]
have
considered
the
308
maximum
number
of
limit
cycles
for
a
dynamical
system
of
Lienard
type:
dxjdt
=
y-
F(x),
dyjdt
=-
g(x),
with
F
and
g
are
polynomials.
They
have
found
that
for
several
classes
of
such
systems,
the
maximum
number
of
limit
cycles
that
can
bifurcate
out
of
a
critical
point
under
perturbation
of
the
coefficients
in
F
and
g
can
be
represented
in
terms
of
the
degree
of
F
and
g.
In
their
classic
paper
Li
and
Yorke
[1975)
posed
a
question
that
for
some
nice
class
of
functions
whether
the
existence
of
an
asymptotically
stable
periodic
point
implies
that
almost
every
point
is
asymptotically
periodic.
Recently,
Nusse
[1987]
established
the
following
two
theorems
which
affirm
the
question
posed
by
Li
and
Yorke
in
part.
Theorem
6.
7.
2
Let
f
be
a
mapping
of
c
1
•
11
for
some
positive
real
number
a,
from
a
nontrivial
interval
X
into
itself.
Assume
that
f
satisfies
the
following
conditions:
(i)
The
set
of
asymptotically
stable
periodic
points
for
f
is
compact
(if
this
set
is
empty,
then
there
exists
at
least
one
absorbing
boundary
point
of
X
for
f).
(ii)
The
set
of
points,
whose
orbits
do
not
converge
to
an
asymptotically
stable
periodic
orbit
of
f
or
converge
to
an
absorbing
boundary
point
of
X
for
f,
is
a
nonempty
compact
set,
and
f
is
an
expanding
map
on
this
set.
Then
we
have:
(a)
The
set
of
points
whose
orbits
do
not
converge
to
an
asymptotically
stable
periodic
orbit
of
f
or
to
an
absorbing
boundary
point
of
X
for
f,
has
Lebesgue
measure
zero.
(b)
There
exists
a
positive
integer
p
such
that
almost
every
point
in
X
is
asymptotically
stable
periodic
with
period
p,
provided
that
f(X)
is
bounded.
As a
consequence,
the
set
of
aperiodic
points
for
f,
or
equivalently,
the
set
on
which
the
dynamical
behavior
of
f
is
chaotic,
has
Lebesgue
measure
zero.
Furthermore,
one
can
show
that
the
conditions
in
Theorem
6.7.2
are
invariant
under
the
conjugation
with
a
diffeomorphism.
Nonetheless,
since
condition
(i)
and
part
of
(ii)
can
not
be
ascertained
a
priori,
they
are
not
very
useful
for
practical
purposes.
309
Theorem
6.7.3
Assume
that
f
is
a
chaotic
c
3
-mapping
from
a
nontrivial
interval
X
into
itself
satisfying
the
following
conditions:
(i)
f
has
a
nonpositive
Schwarzian
derivative,
i.e.,
(d'f(x)jdx')/(df(x)jdx)
-
(3/2)[(d'f(x)jdx')/(df(x)jdx)]'
S
0
for
all
x e X
with
df(x)jdx
+
o:
(ii)
The
set
of
points,
whose
orbits
do
not
converge
to
an
absorbing
boundary
point(s)
of
X
for
f,
is
a
nonempty
compact
set:
(iii)
The
orbit
of
each
critical
point
for
f
converges
to
an
asymptotically
stable
periodic
orbit
of
f
or
to
an
absorbing
boundary
point(s)
of
X
for
f:
(iv)
The
fixed
points
of
f'
are
isolated.
Then
we
have:
(a)
The
set
of
points
whose
orbits
do
not
converge
to
an
asymptotically
stable
periodic
orbit
of
f
or
to
an
absorbing
boundary
point(s)
of
X
for
f,
has
Lebesgue
measure
zero:
(b)
There
exists
a
positive
integer
p
such
that
almost
every
point
in
X
is
asymptotically
periodic
with
period
p,
provided
that
f(X)
is
bounded.
In
the
following,
we
will
give
some
simple
examples:
(A)
X=
[-1,1],
f:
X~
X
is
defined
by
f(x)
=
3.701x
3
-
2.701x.
It
can
be
shown
that
f
has
two
asymptotically
stable
periodic
orbits
with
period
three.
Since
f
has
a
negative
Schwarzian
derivative
and
dfjdx(x=1)
=
dfjdx(x=-1)
>
1,
by
Theorem
6.7.3
we
have
that
almost
every
point
in
X
is
asymptotically
periodic
with
period
three.
(B)
Let
f
be
a
chaotic
map
of
c
3
from
a
compact
interval
[a,b)
into
itself
with
the
following
properties
[Collet
and
Eckmann
1980):
(i)
f
has
one
critical
point
c
which
is
nondegenerate,
f
is
strictly
increasing
on
[a,c)
and
strictly
decreasing
on
[c,b]:
(ii)
f
has
negative
Schwarzian
derivative:
(iii)
the
orbit
of
c
converges
to
an
asymptotically
stable
periodic
orbit
of
f
with
smallest
period
p,
for
some
positive
integer
p.
But
since
the
existence
of
an
asymptotically
stable
fixed
point
in
[a,f'
(c))
has
not
been
excluded,
thus
the
following
cases
can
occur:
(1)
f
has
an
asymptotically
stable
fixed
point
in
310
[a,f•
(c))
and
f
has
an
asymptotically
stable
periodic
orbit
which
contains
the
critical
point
in
its
direct
domain
of
attraction.
(2)
f
has
an
asymptotically
stable
fixed
point
in
[a,f•
(c))
and
the
orbit
of
the
critical
point
converges
to
this
stable
fixed
point.
Moreover
f
has
2"
unstable
periodic
points
with
period
n
for
each
positive
integer
n.
(3)
f
has
no
asymptotically
stable
fixed
point
in
[a,f
2
(c));
consequently,
the
critical
point
is
in
the
direct
domain
of
attraction
of
the
asymptotically
stable
periodic
orbit.
Since
the
map f
satisfies
the
conditions
of
Theorem
6.7.3
and
we
have
that
almost
each
point
in
the
interval
[a,b)
is
asymptotically
periodic
with
period
p.
For
one-dimensional
maps
x~
1
=
mF(x"),
there
are
four
kinds
of
bifurcation
likely
to
take
place
in
general,
namely:
(a)
regular
period
doubling,
(b)
regular
period
halving,
(c)
reversed
period
doubling,
and
(d)
reversed
period
halving.
It
is
known
that
if
the
Schwarzian
derivative
is
negative
at
the
bifurcation
point,
then
either
(a)
or
(b)
can
take
place,
(if
the
Schewarzian
derivative
is
positive,
then
either
(c)
or
(d)
will
take
place)
[Singer
1978,
Guckenheimer
and
Holmes
1983,
Whitley
1983].
Thus,
having
a
negative
Schwarzian
derivative
is
a
condition
which
implies
that
at
the
parameter
value
where
a
period
doubling
bifurcation
occurs,
the
periodic
orbit
is
stable
due
to
nonlinear
terms
and
prevent
(c)
and
(d)
to
occur.
There
were
questions
raised
as
to
whether
or
not
having
a
negative
Schwarzian
derivative
ruled
out
regular
period
halving.
Recently,
Nusse
and
Yorke
[1988]
presented
an
example
of
a
one-dimensional
map
where
F(x)
is
unimodal
and
has
a
negative
Schwarzian
derivative.
They
showed
that
for
their
example,
some
regular
period
halving
bifurcations
do
occur,
and
the
topological
entropy
can
decrease
as
the
parameter
m
is
increased.
So
far
we
have
been
discussing
bifurcations
of
nonlinear
continuous
or
discrete
systems.
George
[1986]
has
demonstrated
that
bifurcations
do
occur
in
a
piecewise
linear
system.
311
Before
we
end
this
section
on
chaos,
we
would
like
to
point
out
that
it
is
fairly
simple
to
demonstrate
deterministic
chaos
experimentally.
Briggs
[1987]
gives
five
simple
nonlinear
physical
systems
to
demonstrate
the
ideas
of
period
doubling,
subharmonics,
noisy
periodicity,
intermittency,
and
chaos
in
a
teaching
laboratory,
such
as
in
senior
or
graduate
laboratory
courses.
So
far
we
have
only
scratched
the
surface
of
chaos,
nonetheless,
we
hope
that
we
have
provided
some
sense
of
the
richness
and
diversity
of
the
phenomenology
of
chaotic
behavior
of
dynamical
systems.
It
will
remain
to
be
an
exciting
field
for
sometime
to
come,
because
not
only
there
will
be
new
phenomena
to
be
discovered,
but
more
importantly,
there
will
be
many
surprises
along
the
way.
For
instance,
Grebogi,
Ott
and
Yorke
[1982]
have
investigated
the
chaotic
behavior
of
a
dynamical
system
at
parameter
values
where
an
attractor
collides
with
an
unstable
periodic
orbit,
specifically,
the
attractor
is
completely
annihilated
as
well
as
its
basin
of
attraction.
They
call
such
events
crises,
and
they
found
sudden
qualitative
changes
taken
place
and
the
chaotic
region
can
suddenly
widen
or
disappear.
It
is
well-known
that
in
periodically
driven
systems,
the
intersection
of
stable
and
unstable
manifolds
of
saddle
orbits
forms
two
topologically
distinct
horseshoes,
and
the
first
one
is
associated
with
the
destruction
of
a
chaotic
attractor,
while
the
other
one
creates
a new
chaotic
attractor.
Abrupt
annihilation
of
chaotic
attractors
has
been
observed
experimentally
in
a
driven
C0
2
laser
and
p-n
junction
circuits.
Recently,
Schwartz
[1988]
used
a
driven
C0
2
laser
model
to
illustrate
how
sequential
horseshoe
formation
controls
the
birth
and
death
of
chaotic
attractors.
Hilborn
[1985)
reported
the
quantitative
measurements
of
the
parameter
dependence
of
the
onset
of
an
interior
crisis
for
a
priodically
driven
nonlinear
diode-inductor
circuit.
The
measurements
are
in
reasonable
agreement
with
the
predictions
of
Grebogi,
Ott
and
Yorke
[1982).
Grebogi,
ott
and
Yorke
[1986]
recently
312
have
also
found
a
theory
of
the
average
lifetime
of
a
crisis
for
two-
dimensional
maps.
Recently,
Yamaguchi
and
Sakai
[1988]
studied
the
structure
of
basin
boundaries
for
a
two-dimensional
map
analytically,
and
they
have
shown
that
basin
boundaries
do
not
play
the
role
of
a
barrier
when
the
crisis
occurs
as
the
two
parameters
are
changed.
The
studies
of
nonlinear
dynamical
systems
or
maps
modeling
these
systems
have
been
concerned
with
how
chaos
arises
through
successive
series
of
instabilities.
The
converse
questions
are:
is
there
a
nonlinear
system
or
map
whereby
at
certain
control
parameters
the
dynamics
of
the
system
can
lead
from
chaotic
states
to
ordered
states?
or
how
can
the
interaction
of
chaotic
attractors
lead
to
ordered
states?
Recently,
Phillipson
[1988]
has
demonstrated
the
emergence
of
ordered
states
out
of
a
chaotic
background
of
states
by
suitably
contrived
one-dimensional
maps
characterized
by
multiple
critical
points.
Certainly
similar
considerations
of
maps
of
higher
dimensionality
will
be
investigated.
Bolotin,
Gonchar,
Tarasov
and
Chekanov
[1989)
recently
have
studied
the
correlations
between
the
characteristic
properties
of
the
classical
motion
and
statistical
properties
of
energy
spectra
for
two-dimensional
Hamiltonian
with
a
localized
region
having
a
negative
Gaussian
curvature
of
the
potential
energy
surface.
They
have
found
that
at
such
potential
energy
surface
with
negative
Gaussian
curvature,
the
transition
of
regularity-chaos-regularity
occurs.
Further
studies
of
a
more
general
nature
will
be
of
great
interest.
It
should
be
pointed
out
that
the
establishment
of
order
out
of
chaos
implies
the
imposition
of
rules
on
an
otherwise
unspecified
situation.
This
is
the
basis
of
evolutionary
processes
in
nature
[Farmer,
Lapedes,
Packard
and
Wendroff
1986].
One
can
easily
show
or
construct
nonlinear
dynamical
systems
with
multiple
attractors
and
consequently
require
very
accurate
initial
conditions
for
a
reliable
prediction
of
the
final
states.
This
is
the
issue
raised
by
Grebogi,
McDonald,
Ott
and
Yorke
[1983].
We
will
leave
this
section
313