Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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structures.
And
the
cubic
autocatalytic
system
also
provides
the
simplest,
yet
chemically
consistent,
example
of
temporal
and
spatial
oscillations
in
a
reaction-diffusion
system.
We
would
like
to
mention
that
the
existence
of
two
stable
solutions
on
S
is
a common
and
well-known
feature
of
higher
order
autocatalyic
reactions.
It
is
also
striking
to
note
that
higher
order
autocatalytic
reaction
networks
are
very
sensitive
on
initial
conditions.
In
the
following
paragraphs,
we
shall
briefly
discuss
competition
between
higher
order
autocatalytic
reactions.
As
in
Eq.(7.5-9),
we
have
A + 2X;
c~··c;
3X;r
I;
d·'
..
d;
B,
B r
....
A,
i E
[l,n].
For
simplicity,
let
us
j~st
discuss
the
case
with
competitors,
i.e.,
n =
2,
and
irreversible
degradation
(d;'=
0).
Then
the
dynamical
equations
are:
dbjdt
= d
1
x
1
+ d
2
x
2
-
rb,
dx
1
jdt
= x
1
(c
1
ax
1
-
c
1
•x
1
2
dx
2
jdt
= x
2
(c
2
ax
2
-
c
2
•x
2
2
with
a = c
0
- x
1
-
x
2
-
b.
(7.
5-15a)
(7.
5-15b)
(7.5-15c)
Again,
the
fixed
points
P
0
(x
1
= x
2
b =
0)
is
stable.
Furthermore,
any
X;
which
is
very
small
is
characterized
by
dx;/dt
<
0,
thus
will
tend
toward
zero.
Contrary
to
the
competition
between
linear
autocatalytic
system
Eq.(7.5-10),
all
invariant
surfaces
of
S
are
attracting.
On
both
planes,
i.e.,
x
1
= 0
and
x
2
=
0,
we
observe
all
the
behavior
discussed
earlier.
For
instance,
we
obtain
two
stable
fixed
points
P
1
(x
1
> o, x
2
=
0)
and
P
2
(x
1
= o, x
2
>
0).
Increasing
c
0
further
for
c;'>
o,
we
observe
the
creation
of
two
pairs
of
fixed
points
in
the
interior
of
s.
If
c
0
is
large
enough,
one
of
the
fixed
point
P
12
(x
1
> o, x
2
>
0)
will
be
stable.
Nonetheless,
the
various
bifurcations
occurring
with
increasing
c
0
are
diffcult
to
analyze
and
one
may
expect
chaotic
behavior
near
the
Hopf
and
homoclinic
bifurcations.
In
the
case
of
c;'
=
0,
only
one
pair
of
fixed
points
emerges
in
the
interior
of
s.
These
are
always
a
source
and
a
saddle
point.
There
is
no
stable
fixed
point
in
S
even
for
large
c
0
•
This
indicates
that
in
the
irreversible
case,
all
374
orbits
converge
to
one
of
the
planes
x
1
= 0
or
x
2
= o.
Thus,
at
least
one
of
the
competitors
has
to
die
out,
just
as
we
have
found
in
the
linear
case,
there
is
no
region
of
coexistence
of
the
two
competitors
when
the
autocatalytic
processes
are
irreversible.
For
further
discussion
on
autocatalysis,
hypercycles,
and
permanence,
see
for
instance,
the
papers
in
Schuster
[1984],
Freedman
and
Waltman
[1977],
Epstein
[1979],
Eigen
et
al
[1980].
For
more
quantitative
results
concerning
competitions
between
three
species,
see
the
classical
paper
by
May
and
Leonard
[1975).
Even
though
the
predator-prey
model
of
interacting
populations
in
terms
of
Lotka-Volterra
equations
and
the
autocatalysis
equations
in
terms
of
reaction-diffusion
equations
are
mainly
for
the
study
of
population
ecology
and
biochemical
reactions,
it
should
be
pointed
out
that
the
laser
equations
for
a
two-level
system
have
the
exactly
same
set
of
differential
equations
as
the
original
Lotka-Volterra
equations.
Likewise,
some
problems
in
semiconductor
physics
have
very
similar
differential
equations
as
the
Lotka-Volterra
equations
or
the
autocatalysis
equations
we
shall
see
in
the
next
section.
7.6
Examples
in
semiconductor
physics
and
semiconductor
lasers
By
looking
at
the
dynamical
equations
of
the
predator-prey
problem
with
crowding
and
autocatalysis,
one
can
easily
recognize
their
formal
similarities
or
equivalence
to
some
of
the
problems
in
semiconductor
physics
or
semiconductor
laser
physics.
In
this
section,
we
want
to
show
that
the
formalism
and
results
obtained
in
Section
5
can
readily
be
applied
to
some
situations
in
semiconductor
physics
and
semiconductor
lasers.
(i)
Limit
cycle
for
excitons:
There
has
been
some
work
on
the
possibility
of
375
spontaneous
oscillations
of
the
electron
concentration
and
temperature
due
to
nonlinear
dependence
of
the
Auger
coefficient
on
temperature
[Degtyarenko,
Elesin
and
Furmenov
1974].
Nonetheless,
this
model
does
not
contain
an
autocatalytic
process,
here
we
consider
such
process
following
Landsberg
and
Pimpale
[1976]
and
Pimpale
et
al
[1981)
in
a
model
with
the
following
three
steps:
stimulated
production
of
excitons:
e + h + X
c~
2x,
radioactivedecay
of
excitons
at
recombination
center:
X
d~
r,
photogeneration
of
carriers:
r
~
e +
h.
It
is
very
clear
that
these
processes
are
irreversible
first
order
autocatalytic
processes
as
in
Eq.(7.5-3)
where
(e
+h)
=A,
r =
B,
c'=
d'=
o.
The
irreversibility
is
because
the
semiconductor
is
assumed
to
be
far
from
equilibrium.
Here
we
assume
that
the
semiconductor
to
be
nondegenerate
and
under
high
excitation
so
that
the
electron
and
hole
concentrations
are
taken
to
be
equal.
Then
the
differential
equations
describing
the
dynamical
system
are
(one
can
obtain
from
Eq.(7.5-4)
too):
dnjdt
=
gb-
cn•x,
dxjdt
=
cn•x-
dx.
(7.6-1a)
(7.6-1b)
The
conservation
law
of
total
concentration
and
the
other
dynamic
equation
(reduced
form
of
Eq.(7.5-4b))
are
taken
into
account.
Note
that,
the
exciton
decay
process
has
interesting
nonlinear
features:
the
decay
rate
is
proportional
to
the
number
of
recombination
centers
available
where
there
is
a
large
number
of
excitons
available
for
decay,
i.e.,
for
x >>
1/q,
dxjdt
~
-1/q
where
1/q
is
concentration
of
recombination
centers.
But
when
the
number
of
excitons
is
much
lower
than
the
number
of
recombination
centers,
the
exciton
decay
rate
is
proportional
to
the
exciton
concentration,
i.e.,
for
x
<<
1/q,
dxjdt
~
-
x.
Since
the
Michaelis-Menten
S-shaped
law
seems
to
be
a
good
interpolation
law
for
the
exciton
decay
rate,
we
then
replace
Eq.(7.6-1b)
by
dx/dt
=
cn•x-
dx/(1
+
qx).
(7.6-1b')
The
exciton
decay
rate
on
recombination
centers,
given
by
376
the
second
term
of
Eq.(7.6-1b'),
is
shown
qualitatively
in
Fig.7.6.1.
dx
L----------------------------------X
Fig.7.6.1
For
small
x,
(xq
<<1),
one
has
the
usual
exponential
decay.
But
for
large
x,
there
is
a
departure
from
the
usual
exponential
decay.
Experimentally,
such
a
departure
has
been
observed
in
qualitative
agreement
with
Fig.
7.6.2
[Klingenstein
and
Schmod
1979].
A
generalization
to
an
n-th
order
reaction
is
also
possible
[Ibanez
and
Velarde
1977],
but
we
shall
not
discuss
it
any
further.
Note,
here
we
have
also
ignored
electron-hole,
electron-impurity
recombinations
and
other
excitonic
processes.
c
1
4
~-------------------------------
G
(0,0)
Fig.7.6.2
377
We
can
rescale
the
dimensions
in
such
a
way,
(as
an
exercise)
such
that
Eq.(7.6-1a,b')
becomes
dnjdt
=
G-
cn•x,
dxjdt
=
cnx•
-
X/(1
+
x).
(7.6-2)
For
physically
meaningful
solutions,
we
have
the
following
constraints:
X
~
0,
n
~
O, G > O, c >
0.
As
in
section
7.5-(ii),
there
are
two
fixed
points:
P
0
( x = 0 , n =
oo)
and
P
1
(
x = x
0
,
n = n
0
)
where
x
0
=
G/
( 1 -
G)
, n
0
= I ( 1 -
G)
I c I
y,
and
0 < G <
1.
Clearly,
P
0
is
unstable,
by
noting
that
changing
variable
y
=
1/n,
P
0
(X
=
0,
y =
0)
is
unstable
with
respect
to
even
linear
perturbation.
To
get
a
feel
of
the
stability
of
P
1
,
let
x(t)
= x
0
+
u(t),
n(t)
= n
0
+
v(t),
then
Eq.~;~:-2~~]b":o~~·[~]
•
where
G(1
-
G)
2G(c/(1
-
G)
)y,
J
A=
-(1-
G)
-2G(c/(1-
G))y,
(7.6-3)
Note
that
det
A =
2G(1
-
G)
312
cy,
> 0
for
physically
meaningful
values
of
G
and
c.
And,
the
Poincare
index
of
the
fixed
points
is
+1.
Thus
the
stability
of
P
1
is
determined
by
the
value
of
Tr
A,
where
Tr
A=
G(1
-
G)
-
2[c/(1
-
G)]Y..
The
bifurcation
values
of
G
and
care
given
by
Tr
A=
0,
i.e.,
(1-
G)
3
=
4c.
Since
(0,0)
is
unstable,
it
appears
that
the
fixed
point
is
stable
for
Tr
A < o
and
unstable
for
Tr
A >
0.
The
unstable
steady
state
leads
to
oscillation.
See
Fig.7.6.2.
Another
interesting
phenomenon
is
the
Gunn
effect,
or
Gunn
instability
[see
for
instance,
Seeger
1982]
in
a
semiconductor
having
a
two-valley
conduction
band,
such
as
GaAs,
is
caused
by
the
negative
resistance
arising
from
the
transfer
of
electrons
in
high
fields
from
a
lower
light-mass
valley
to
an
upper
heavy-mass
valley.
It
has
been
determined
that
the
Gunn
instability
gives
rise
to
anomalously
large
low-frequency
current
noise
over
a
wide
range
of
fields
beyond
the
threshold
of
the
instability.
Recently,
Nakamura
378
[1988]
has
performed
carrier
dynamics
simulation
to
investigate
the
low-frequency
current
noise
mechanism
in
a
two-valley
semiconductor.
It
has
been
noted
that
the
current
noise
is
caused
by
the
chaotic
transfer
of
carriers
in
the
accumulation
layer
of
a
high-field
domain.
7.7
control
systems
with
delayed
feedback
Roughly
speaking,
the
purpose
of
control
is
to
manipulate
the
available
inputs
of
a
dynamical
system
to
cause
the
system
to
behave
in
a
more
desirable
manner
than
the
one
without
control.
Almost
every
movement
our
body
makes
in
our
daily
life
is
under
control.
In
fact,
most
of
them
involve
feedback
of
various
means.
This
can
also
easily
generalize
to
our
physical
world.
There
are
two
major
types
of
control.
In
open-loop
control,
the
input
is
generated
by
some
processes
external
to
the
system
itself,
and
then
is
applied
to
the
system.
A
specific
input
may
be
generated
by
analysis
such
as
developing
the
yearly
production
plan
for
a
company,
or
may
be
generated
repeatedly
by
a
physical
device
when
directing
a
physical
parocess
or
machine
such
as
programmable
switch
on
and
off
of
a
heating
system.
At
any
rate,
the
underlying
feature
of
open-loop
control
is
that
the
input
function
is
determined
completely
by
an
external
process.
In
closed-loop
control,
the
input
is
determined
on
a
continuing
basis
by
the
behavior
of
the
system
itself
as
expressed
by
the
behavior
of
the
outputs.
This
is
also
called
feedback
control
because
the
outputs
are
fed
back
to
the
input.
There
are
many
reasons
why
feedback
control
is
often
preferable
to
open-loop
control.
One
is
that
a
feedback
rule
is
often
simpler
then
a
comparable
open-loop
scheme
because
it
may
require
a
fair
amount
of
computation
and
complex
implementation.
It
is
very
important
that
feedback
can
automatically
adjust
to
unforeseen
system
changes
or
to
unanticipated
input
disturbances.
One
of
the
other
important
reasons
is
that
much
feedback
can
rapidly
adjust
to
changes
379
leads
to
the
fact
that
feedback
can
increase
the
stability
of
the
system.
In
this
respect,
mathematically,
a
basic
feature
of
feedback
is
that
it
can
influence
the
characteristic
polynomial
of
the
system.
Let
us
consider
the
effect
of
feedback
applied
to
a
very
simple
system,
dx(t)/dt
=
u(t).
One
can
easily
show
that
this
system
is
marginally
stable.
Now
suppose
the
input
is
controlled
by
feedback
and
suppose
u(t)
=
a(x
0
-
x(t)).
This
system
certainly
yields
x(t)
~
x
0
•
Note
that
by
defining
y
x - x
0
,
the
feedback
system
is
governed
by
the
simple
equation
dy(t)jdt
=
-ay(t),
and
the
variable
y(t)
is
the
error.
Thus,
feedback
has
converted
the
original
marginally
stable
linear
system
into
an
asymptotically
stable
system.
As
stated
before,
an
important
objective
of
feedback
is
to
make
the
system
more
stable
than
it
would
otherwise
be.
Therefore,
it
is
natural
to
ask
how
much
influence
feedback
can
have
on
the
eigenvalues
of
a
system.
An
important
result
is
the
eigenvalue
placement
theorem.
This
theorem
states
that
if
a
system
is
completely
controllable
and
if
all
state
variables
are
available
as
outputs,
then
by
suitable
direct
feedback
it
is
possible
for
the
feedback
system
to
have
any
desired
characteristic
polynomial.
Before
we
continue
on
nonlinear
feedback
control
systems,
we
would
like
to
introduce
the
result
of
Sontag
[1982]
which
deals
with
the
necessary
and
sufficient
conditions
for
asymptotic
controllability
of
a
controllable
system.
The
system
under
consideration
is
of
the
following
type:
dx(t)jdt
=
f(x(t),u(t)),
(7.7-1)
where
the
states
x(t)
£ R",
control
parameter
u(t)
is
in
a
metric
space
X,
and
the
map
f
is
assumed
locally
Liptchitz
to
ensure
the
existence
and
uniqueness
of
solutions
for
small
time
intervals.
Furthermore,
admissible
control
functions
u(t)
are
all
measurable
and
locally
bounded
functions,
and
X
is
topologized
by
the
weak
topology.
The
system
Eq.(7.7-1)
is
asymptotically
controllable
if
the
following
conditions
are
satisfied:
(i)
for
each
state
x
380
there
exists
a
control
for
which
the
corresponding
solution
¢(x,u,t)
is
defined
for
all
t,
and
converges
to
zero;
(ii)
if
for
a
given
e >
o,
there
is
a o > 0
such
that
for
any
x
of
norm
less
than
&,
there
is
a u
as
in
(i)
with
the
ensuing
orbit
having
llx(t)ll
< e
for
all
t;
(iii)
there
is
a
neighborhood
U
of
the
zero
state,
and
a
compact
subset
K
of
X
such
that
if
x(O)
is
in
u,
there
exists
an
input
as
in
(ii)
with
values
in
K
almost
everywhere.
Then
we
have
the
following
theorem:
Theorem
7.7.1
The
system,
Eq.(7.7-1),
is
asymptotically
controllable
iff
there
is
a
Liapunov-like
function
for
it.
Here
a
Liapunov-like
function
for
Eq.(7.7-1)
is
a
real
function
v
defined
on
R"
such
that:
(i)
V
is
continuous;
(ii)
V(x)
> 0
for
X +
O,
V(O) =
0;
(iii)
{xi
V(x)
< a}
is
bounded
for
all
a;
(iv)
for
each
x + 0
there
is
a
relaxed
control
u
with
V'
(x,u)
<
O;
(v)
there
is
a
neighborhood
L
of
the
zero
state
and
a
compact
subset
K
of
U
such
that,
for
states
x
in
L,
the
control
u
can
be
chosen
with
values
in
K
(more
precisely,
the
measures
u(t)
are
supported
inK
almost
everywhere).
The
derivative
along
the
chosen
trajectory
is
defined
as:
V'
(x,u)
=
limt~o
inf[
(V(¢(t,x,u))
-
V(x)
)/t].
In
practice,
the
value
of
u
is
determined
after
a
measurement
on
the
state
x(t)
at
t
was
made.
If
this
measurement
is
in
error,
say
x(t)
+
e(t)
is
measured
rather
than
x(t),
the
governing
equation
of
motion
will
have
the
form
dx(t)/dt
=
X(x(t)
+
e(t)).
(7.7-2)
It
is
this
concept
which
leads
to
the
definition
of
stability
with
respect
to
measurement.
Intuitively,
X
is
stable
with
respect
to
measurement
if
any
solutions
of
Eq.(7.7-2)
and
dx(t)/dt
=
X(x(t)),
satisfying
the
same
initial
conditions,
remain
arbitrarily
close
over
any
finite
positive
time
interval
whenever
the
supremum
of
le(t)
I
over
this
time
interval
is
restricted
to
be
sufficiently
small.
381
In
general,
the
initial
value
problem
for
such
a
vector
field
X may
not
have
a
solution.
Nonetheless,
if
there
is
an
absolutely
continuous
function
~(t)
which
satisfies
the
initial
condition
and
d~(t)/dt
=
X(~(t))
almost
everywhere,
then
~(t)
is
called
a
classical
solution.
Consider
a
control
system
of
the
form
dxjdt
=
g(x,
u(x)),
x =
(x
1
,
x
2
,
•••
,xn),
u =
(u
1
,
•••
,
un)
,
(7.7-3)
with
values
u(x)
to
be
chosen
from
a
control
set
u.
Let
the
target
manifold
s
be
contained
in
[O,~)xR".
If
g
is
bounded
and
Lipschitzian
and
u
is
a
given
Lipschitzian
control,
then
an
initial
value
problem
of
Eq.(7.7-3)
with
data
x(O)
= x
0
has
a
unique
solution,
with
value
at
time
t
denoted
by
~(t,O,x
0
).
Assume
that
~(t,O,x
0
)
£
s.
Then
the
question
is:
If
s
has
dimension
less
than
n
in
R~
1
,
is
it
possible
that,
for
each
x
in
some
neighborhood
N(x
0
)
c
R"
of
x
0
,
there
exists
a
value
t(x),
and
0
~
t(x)
<
~.
such
that
~(t(x),
o,
x)
£
S?
Before
we
address
this
question,
we
shall
briefly
discuss
the
generalized
concept
of
solutions
for
such
equations
as
given
by
Filippov
[1964].
Let
X
be
a
measurable
function
defined
almost
everywhere
in
a
domain
Q c
R"
with
values
in
a
bounded
set
in
R".
With
X(x)
associate
the
convex
set
K{X(x)} = n
6
>
0
nfl<N>=O
closed
convex
hull{X(U(x,
6') - N},
where
U(x,o)
is
a
closed
c5
neighborhood
of
x,
N
an
arbitrary
set
in
R"
and
~
is
n
dimensional
Lebesgue
measure.
Then
an
absolutely
continuous
vector
valued
function
~.
defined
on
[O,T],
is
called
a
Filippov
solution
of
dxjdt
= X(x)
if
for
almost
all
t,
d~(t)/dt
£
K{X(~(t))}.
In
Filippov
[1964],
it
has
been
shown
that
such
solutions
will
always
exist,
and
many
of
their
properties
are
discussed.
In
particular,
if
X
is
continuous,
then
K{X(x)} =
X(x).
Let
U(x,o)
denote
a
compact,
spherical
neighborhood
of
radius
o,
about
the
point
x
in
R",
and
coA
denotes
the
convex
hull
of
a
set
A. A
vector
field
X,
for
which
a
classical
solution
~
of
dxjdt
= X(x)
with
arbitrary
initial
382
data
x
0
exists,
is
said
to
be
stable
with
respect
to
measurement
if
given
£ > 0
and
finite
T > o,
there
exists
a
6 > 0
such
that
whenever
e
is
a
measurable
function
on
[O,T]
with
values
in
R"
and
norm
less
than
6
for
which
a
corresponding
solution~
of
dxjdt
=
X(x(t)
+
e(t)),
x(O)
x
0
,
exists
on
[O,T],
then
lltt>
-
~11
<
£.
Hermes
[1967]
has
established
the
following
theorem:
Theorem
7.7.2
If
X
is
stable
with
respect
to
measurement,
then
every
classical
solution
is
a
Filippov
solution.
If
X
is
stable
with
respect
to
measurement,
solutions
for
t
~
0
of
the
initial
value
problem
for
the
corresponding
differential
equation
are
unique,
and
such
a
solution,
when
evaluated
at
a
fixed
positive
time,
varies
continuously
with
the
initial
data.
Thus,
with
increasing
time,
solutions
may
join
but
not
branch.
Thus,
it
is
felt
that
feedback
controls
which
are
meaningful
from
the
standpoint
of
applications
should
lead
to
vector
fields
which
are
stable
with
respect
to
measurement.
Nonetheless,
to
characterize
such
vector
fields
directly
is
not
a
easy
task.
In
the
next
section,
we
shall
illustrate
such
a
system
in
feedback
control
of
semiconductor
lasers
and
phased
arrays.
For
linear
difference-differential
equations
of
retarded
delay
feedback,
Infante
[1982]
has
outlined
a
characterization
of
a
broad
class
of
such
difference-
differential
systems
which
have
the
property
that
their
asymptotic
stability
and
hyperbolicity
characteristics
are
not
affected
by
the
value
of
the
delays
involved.
Recently,
an
der
Heiden
and
Walther
[1983]
and
an
der
Heiden
[1983]
have
shown
that
under
centain
conditions,
the
difference-differential
equation
dx(t)/dt
=
f(x(t-1))
-
ax(t),
(7.
7-4)
describing
delayed
feedback
control
systems,
admits
chaotic
solutions.
Recurrent
synaptic
feedback
is
an
important
mechanism
for
regulating
synchronous
discharges
of
a
population
of
neurons.
It
can
be
found
in
the
brain,
the the
spinal
chord
383