Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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the
differential
equations,
drjdt
=
r(1-
r),
d6jdt
=
sin
2
(6/2).
The
orbits
are
shown
in
Fig.5.1.2.
These
consist
of
two
fixed
points
p
1
=
(0,0)
and
p
2
=
(1,0),
an
orbit
ron
the
unit
circle
with
{p
2
}
asd
the
positive
and
the
negative
limit
set
of
all
points
on
the
unit
circle.
All
points
p,
where
p + p
2
,
have
n(p)
=
{p
2
}.
Thus
{p
2
}
is
an
attractor.
But
it
is
neither
stable
nor
a
uniform
attractor.
Note
also
that
for
any
p =
(a,O),
a>
0,
J+(p)
is
the
unit
circle.
Fig.5.1.2
Now
let
us
state
some
results
about
attractors.
Theorem
5.1.3
If
M
is
a
weak
attractor, attractor,
or
uniform
attractor,
then
the
coresponding
set
Aw(M),
A(M),
or
Au(M)
is
open
(indeed
an
open
neighborhood
of
M).
Theorem
5.1.4
If
M
is
stable
then
it
is
positively
invariant.
As
a
consequence,
if
M =
{x},
then
xis
a
fixed
point.
Theorem
5.1.5
A
set
is
stable
iff
o•(M) =
M.
In
view
of
the
above
theorem,
we
can
give
the
following
204
definition.
If
a
given
set
M
is
unstable,
the
non-empty
set
o•(M) - M
will
be
called
the
region
of
instabilitv
of
M.
If
o•(M)
is
not
compact,
then
M
is
said
to
be
globally
unstable.
Theorem
5.1.6
If
M
is
stable
and
is
a
weak
attractor,
then
M
is
an
attractor
and
consequently
asymptotically
stable.
Theorem
5.1.7
If
M
is
positively
invariant
and
a
uniform
attractor,
then
M
is
stable.
Consequently
M
is
asymptotically
stable.
Theorem
5.1.8
If
M
is
asymptotically
stable
then
M
is
a
uniform
attractor.
Theorem
5.1.9
Let
f(t,x)
in
dxjdt
=
f(t,x)
be
independent
of
t
or
periodic
in
t.
Then,
stability
of
the
origin
implies
uniform
stability,
and
asymptotic
stability
implies
unform
asymptotic
stability
[Yoshizawa
1966].
Note
that
there
is
no
such
theorem
for
attractivity
or
uniform
attractivity.
The
next
few
theorems
determine
whether
the
components
of
a
stable
or
asymptotically
stable
set
inherit
the
same
properties.
We
would
like
to
point
out
(it
is
very
easy
to
show)
that
in
general
the
properties
of
weak
attraction,
attraction,
and
uniform
attraction
are
not
inherited
by
the
components.
Theorem
5.1.10
A
set
M
is
stable
iff
every
component
of
M
is
stable.
It
should
be
noted
that
the
above
theorem
holds
even
if
X
is
not
locally
compact
[Bhatia
1970].
Theorem
5.1.11
Let
M
be
asymptotically
stable,
and
let
M*
be
a
component
of
M.
Then
M*
is
asymptotically
stable
iff
it
is
an
isolated
component.
The
following
very
important
theorem
is
a
corollary
to
the
above
theorem.
Theorem
5.1.12
Let
X
be
locally
compact
and
locally
connected.
Let
M
be
asymptotically
stable.
Then
M
has
a
finite
number
of
components,
each
of
which
is
asymptotically
stable.
Outline
of
proof.
A(M)
is
open
and
invariant.
Since
X
is
205
locally
connected,
each
component
of
A(M)
is
also
open.
The
components
of
A(M)
are
therefore
an
open
cover
for
the
compact
set
M.
Thus
only
a
finite
number
of
components
of
A(M)
is
needed
to
cover
M.
Now
it
is
ealy
to
show
that
each
component
of
A(M)
contains
only
one
component
of
M.
Theorem
5.1.13
Let
M
be
a
compact
weak
attractor.
Then
o•(M)
is
a
compact
asymptotically
stable
set
with
A(D+(M)) =
Aw(M).
Moreover,
o•(M)
is
the
smallest
asymptotically
stable
set
containing
M.
5.2
Asymptotic
stability
and
Liapunov•s
theorem
In
his
Memoire,
Liapunov
[1892]
gave
several
theorems
dealing
directly
with
the
stability
problems
of
dynamical
systems.
His
methods
were
inspired
by
Dirichlet's
proof
of
Lagrange's
theorem
on
the
stability
of
equilibrium,
is
referred
to
by
Russian
authors
as
Liapunov's
second
method.
Liapunov•s
first
method
for
the
study
of
stability
rests
on
considering
some
explicit
representation
of
the
solutions,
particularly
by
infinite
series.
What
is
known
as
Liapunov's
second
(or
direct)
method
is
making
essential
use
of
auxiliary
functions,
also
known
as
Liapunov
functions.
The
simplest
type
of
such
functons
are
c
1
,
V:
IxD
~
R,
(t,x)
V(t,x)
where
I
and
D
as
before.
Note
that
the
space
X
is
assumed
locally
compact
implicitly.
An
interesting
introduction
to
Liapunov's
direct
method
can
be
found
in
La
Salle
and
Lefschetz
[1961].
A
much
more
complete
and
detailed
treatment
of
stability
as
well
as
attractivity
can
be
found
in
Rouche,
Habets
and
Laloy
[1977].
Many
of
the
material
in
this
chapter
are
coming
from
Rouche,
Habets
and
Laloy
[1977].
If
x(t)
is
a
solution
of
dxjdt
f(t,x),
the
derivatives
of
the
time
function
V'
(t)
=
V(t,x(t))
exists
and
dV'(t)jdt
=
cav;ax)(t,x(t))f(t,x(t))
+
cav;at)(t,x(t)).
(5.2-1)
If
one
introduces
the
function
dV/dt:
IxD
~
R,
(t,x)
~
dV(t,x)jdt
by
dV(t,x)jdt
=
(av;ax)
(t,x)f(t,x)
+
(av;at)
(t,x),
it
then
follows
that
dV'(t)/dt
=
206
dV(t,x(t))/dt.
Thus,
computing
dV'(t)jdt
at
some
given
time
t
does
not
require
a
knowledge
of
the
solution
x(t),
but
only
the
value
of
x
at
t.
Another
useful
type
of
function
is
the
function
of
class
~,
i.e.,
a
function
a €
K,
a:
R+
~
R+,
continuous,
strictly
increasing,
with
a(O)
=
o.
We
call
a
function
V(t,x)
positive
definite
on
D,
if
V(t,x)
is
defined
as
before,
and
V(t,O)
= 0
and
for
some
function
a €
K,
every
(t,x)
€
IxD
such
that
V(t,x)
~
aCIIxll>.
When
we
say
that
V(t,x)
is
positive
definite
without
mentioning
D,
it
means
that
for
some
open
neighborhood
D' c
D
of
the
origin,
Vis
positive
definite
on
D'.
A
well
known
necessary
and
sufficient
condition
for
a
function
V(t,x)
to
be
positive
definite
on
D
is
that
there
exists
a
continuous
function
v*: D
~
R
such
that
v*(o)
=
o,
v*(x)
> o
for
x €
D,
x +
o,
and
furthermore
V(t,O)
= 0
and
V(t,x)
~
v*(x)
for
all
(t,x)
€
IxD.
The
following
two
theorems
are
pertinent
to
the
differential
equation
dxjdt
=
f(t,x).
Theorem
5.2.1
[Liapunov
1892]
If
there
exists
a c
1
function
V:
IxD
~
R
and
for
some a €
K,
and
every
(t,x)
€
IxD
such
that,
(
i)
V (
t,
x)
~
a (
II
x
II
) : V (
t,
o) = o :
(ii)
V(t,x)
S o:
then,
the
origin
is
stable.
Proof:
Let
t
0
€ I
and
€ > 0
be
given.
Since
V
is
continuous
and
V(t
0
,0)
=
O,
there
is
a 6 =
6(t
0
,€)
> 0
such
that
V(t
0
,x
0
)
<
a(€)
for
every
X
0
€ B
6
•
Writing
x(t:t
0
,X
0
)
by
x(t),
and
using
(ii),
one
obtains
for
every
X
0
€ B
6
and
every
t €
J+
: aCIIxll> S
V(t,x(t))
S
V(t
0
,X
0
) <
a(€).
But
since
a €
K,
one
obtains
that
Ux(t)U < € •
Theorem
5.2.2
In
addition
to
the
assumptions
in
Theorem
5.2.1,
if
for
some b € K
and
every
(t,x)
€
IxD:
V(t,x)
S
bCIIxll>,
then
the
origin
is
uniformly
stable.
Let
us
use
this
theorem
to
derive
the
well-known
stability
criteria
for
linear
systems
or
the
linear
approximation.
Let
x
be
an
n-vector
in
R",
and
the
system
can
be
written
as:
207
dxjdt
= Px +
q(x,t),
where
P
is
a
constant
nonsingular
matrix
and
the
nonlinear
term
q
is
quite
small
with
respect
to
x
for
all
t
~
o.
For
simplicity,
without
loss
of
generality,
we
assume
that
the
components
of
q
have
in
some
region
D
and
for
t
~
0
continuous
first
partial
derivatives
in
xk
and
in
t.
Thus,
in
D
and
for
t
~
o,
the
existence
theorem
applies.
Let
us
assume
that
the
characteristic
roots
r
1
,
r
2
,
•••
, rn
of
the
matrix
P
are
all
distinct
and
consider
first
the
case
where
they
are
all
real.
Clearly,
one
can
choose
coordinates
in
such
a way
that
Pis
diagonalized
and
P =
diag(r
1
,
•••
,rn>·
Then,
(a)
if
the
roots
are
all
negative.
Take
V = x,z + . • . +
xn
2
,
and
dV/dt
=
(r
1
x(
+
•••
+ rnxn•) +
s(x,t),
where
s
is
small
compared
with
the
parenthesis.
Thus,
in
a
sufficiently
small
region
D
both
V
and
-dV/dt
are
positive
definite
functions.
Thus
they
satisfy
the
conditions
of
Theorem
5.2.1,
thus
the
origin
is
asymptotically
stable.
(b)
Some
of
the
roots,
say,
r
1
,
•••
, rp
(p
<
n)
are
positive,
and
the
remaining
are
negative.
Now
take
V =
x,'
+ +
xp•
- xp+l - -
xn•
.
Then
we
have,
dV/dt
=
(r
1
x,'
+
•••
+
rPxP
2
-
rp+
1
xp+
1
-
•••
- rnxn•) +
s(x,t)
where
s
is
small
compared
with
the
parenthesis.
At
some
points
arbitrarily
near
the
origin
v
is
positive.
Since
rp+
1
,
•.•
, rn <
0,
thus
dV/dt
is
positive
definite,
and
the
origin
is
unstable.
In
other
words,
a
sufficient
condition
for
the
origin
of
the
system
to
be
asymptotically
stable
is
that
the
characteristic
roots
all
have
negative
real
parts.
If
there
is
a
characteristic
root
with
a
positive
real
part,
then
the
origin
is
unstable.
As
an
example,
there
is
an
interesting
application
to
the
standard
closed
RLC
circuit
with
nonlinear
coupling.
The
equation
of
motion
of
the
charge
q
is:
Ld
2
qjdt•
+
Rdqjdt
+
qjC
+
g(x,dxjdt))
=
o,
where
dqjdt
is
the
current,
g
represents
nonlinear
coupling
with
terms
of
at
least
of
second
order.
We
can
rewrite
this
208
second
order
differential
equation
as:
dqjdt
=
i,
di/dt
=
-qJLC-
iR/L-
g(q,i).
Clearly,
the
origin
is
a
critical
point
and
its
characteristic
roots
are
the
roots
of
R
2
+
Rr/L
+
1/LC
=
o.
since
R,
L
and
c
are
positive,
the
roots
have
negative
real
parts.
If
R
2
/L
2
<
4/LC
or
R
2
<
4L/C,
then
the
characteristic
roots
are
both
complex
with
negative
real
part
-R/L.
When
this
happens,
we
have
spirals
as
paths
and
the
origin
is
asymptotically
stable.
The
origin
is
a
stable
focus.
If
R
2
>
4L/C,
the
origin
is
a
stable
node
and
is
asymptotically
stable.
Next,
we
want
to
introduce
the
concept
of
partial
stability
and
the
corresponding
stability
theorem.
Let
two
integers
m,
and
n >
o,
and
two
continuous
functions
f:
IxDxRm -+ R",
g:
IxDxRm -+
Rm,
where
I = ( 1 ,co) , D
is
a
domain
of
R"
containing
the
origin.
Assume
f(t,o,o)
=
o,
g(t,O,O)
=
0
for
all
t E
I,
and
f
and
g
are
smooth
enough
so
that
through
every
point
of
IxDxRm
there
passes
one
and
only
one
solution
of
the
differential
system
dxjdt
=
f(t,x,y)
,
dyjdt
=
g(t,x,y)
•
(5.2-2)
To
shorten
the
notation,
let
z
be
the
vector
(x,y)
E
R~m
and
z(t;t
0
,Z
0
)
=
(x(t;t
0
,z
0
),y(t;t
0
,z
0
))
for
the
solution
of
the
system.
Eq.5.2-2
starting
from
Z
0
at
t
0
•
The
solution
z = 0
of
Eq.(5.2-2)
is
stable
with
respect
to
x
if
given
E >
0,
and
t
0
E
I,
there
exists
6 > 0
such
that
llx(t;t
0
,z
0
)ll < E
for
all
Z
0
E B
6
and
all
t E
J+.
Uniform
stability
with
respect
to
x
is
defined
accordingly.
Theorem
5.2.3
If
there
exists
a c
1
function
v:
IxDxRm-+
R
such
that
for
some
a E K
and
every
(t,x,y)
E IxDxRm,
(i)
V(t,x,y)
~
aCIIxlll,
V(t,O,O)
=
0;
(ii)
dV(t,x,y)/dt
S
o;
then
the
origin
z = 0
is
stable
with
respect
to
x.
Moreover,
if
for
some
b E K
and
every
(t,x,y)
E IxDxRm,
(iii)
V(t,x,y)
S bCIIxll + IIYIIl;
then
the
origin
is
uniformly
stable
in
x.
Examples:
(i)
Consider
the
linear
differential
equation
dxjdt
=
(D(t)
+
A(t))x,
where
D
and
A
are
nxn
matrices,
209
being
continuous
functions
oft
on
I e
(r,~),
D
diagonal,
A
skew-
symmetric,
x € R".
Choosing
V(t,x)
=
(x,x),
(where
(x,x)
is
a
scalar
product),
we
get
dV(t,x)/dt
=
2(x,D(t)x).
If
the
elements
of
D
are
~
0
for
every
t €
I,
then
V
~
0
and
from
Theorem
5.2.2,
the
origin
is
uniformly
stable.
(ii)
Next,
we
shall
look
at
the
stability
of
steady
rotation
of
a
rigid
body.
Consider
a
rigid
body
with
a
fixed
point
0
in
some
inertial
frame
and
no
external
force
applied.
Let
I,
M,
N
be
the
moments
of
inertia
with
respect
to
o,
and
n
is
the
angular
velocity
in
the
inertial
frame.
The
Euler
equations
of
n,
in
the
principal
axes
of
inertia
n
=
(p,q,r),
are
Idp/dt
Mdqjdt
Ndr/dt
(M
-
N)qr,
(N-
I)rp,
(I
-
M)pq.
(5.2-3)
The
steady
rotations
around
the
first
axis
correspond
to
the
critical
point
p = p
0
,
q = o, r = o.
Define
new
variables,
x = p - p
0
,
y =
q,
z =
r,
the
critical
point
is
shifted
to
the
origin,
and
Equations
(5.2-3)
becomes
dxjdt
(M
-
N)yz/I,
dy/dt
=
(N-
I)
(p
0
+
X)Z/M,
(5.2-4)
dzjdt
=
(I
-
M)
(p
0
+
x)y/N.
If
I
~
M
~
N,
the
steady
rotation
is
around
the
largest
axis
of
the
ellipsoid
of
inertia.
An
auxiliary
function
suitable
for
Liapunov's
theorem
is
V =
M(M-
I)y
2
+
N(N-
I)Z
2
+
[My
2
+
NZ
2
+
I(X
2
+
2xp
0
))
2
(5.2-5)
which
is
a
first
integral,
such
that
dV/dt
=
0.
It
follows
that
the
origin
is
stable
for
Eq.(5.2-4).
Furthermore,
it
is
even
uniformly
stable
since
the
system
is
autonomous.
If
I>
M
~
N,
one
obtains
another
auxillary
function
using
the
first
integral,
V =
M(I-
M)y
2
+
N(I-
N)Z
2
+
[My
2
+
NZ
2
+
I(X
2
+ 2Xp
0
))
2
•
(5.2-6)
Therefore,
the
steady
rotations
of
the
body
around
the
longest
and
shortest
axes
of
its
ellipsoid
of
inertia
are
stable
with
respect
to
p,
q,
r.
Note
that
the
auxiliary
210
functions
Eqs.(5.2-5)
and
(5.2-6)
are
combinations
of
the
integrals
of
energy
and
of
moment
of
mementa.
We
shall
present
a
general
method
for
constructing
such
combinations
shortly.
(iii)
The
third
example
deals
with
the
stability
of
a
glider.
This
may
as
well
be
a
hovering
bird.
First,
suppose
the
plane
of
symmetry
coincides
at
any
moment
with
a
vertical
plane
in
an
inertia
frame.
Let
v
be
the
velocity
of
its
center
of
mass,
and
e
the
angle
between
v
and
horizontal
axis.
The
longitudinal
"axis"
of
the
glider
is
assumed
to
make
a
constant
angle
a
with
v.
Let
m
be
the
mass
of
the
glider
and
g
be
the
gravitational
acceleration.
Let
C
0
(a)
and
CL(a)
be
the
coefficients
of
drag
and
lift
respectively.
The
equations
of
motion
are:
mdvjdt
= -
mgsine
- C
0
(a)v',
mvdejdt
mgcose
+
CL(a)v'.
Letting
V
0
'
= mg/CL, 1 =
gtjv
0
,
y =
vjv
0
,
and
a = C
0
/CL,
we
then
have
transformed
the
equations
into
dyjdr
= -
sine
-
ay'
dejdr
=
(-cose
+
y')/Y·
(5.2-7)
We
have
introduced
a
non-vanishing
drag
for
future
reference
only.
For
the
moment,
let
a =
o,
then
the
above
equations
have
the
critical
points
y
0
=
1,
eo =
2~k
for
k
an
integer,
all
of
them
corresponding
to
one
and
the
same
horizontal
flight
with
constant
velocity.
Let
us
concentrate
on
y
0
=
1,
eo =
o.
One
can
easily
verify
that
V(y,e)
=
(y
3
/3)
-
ycose
+
2/3
is
a
first
integral
for
Equations
(5.2-7)
with
a = o.
In
some
neighborhood
of
(1,0),
V(y,e)
> o
except
V(1,0)
=
o.
Therefore,
if
the
critical
point
(1,0)
is
translated
to
the
origin,
V
expressed
in
the
new
variables
satisfies
the
hypotheses
of
Theorem
5.2.2,
thus
one
can
prove
the
uniform
stability.
For
more
detail,
see
e.g.,
Etkin
[1959].
211
Fig.5.2.1
We
now
go
back
to
the
general
setting
and
state
some
sufficient
conditions
for
instability.
Theorem
5.2.4
[Chetaev
1934]
If
there
exist
t
0
€
I,
€ >
o,
an
open
sets
c
B,,
(with
B;
c
D),
and
a c
1
function
V:
[t
0
,w)xB,
~
R
such
that
on
[t
0
,w)xS:
(i)
0 <
V(t,x)
~
k < w ,
for
some
k;
(ii)
dV(t,x)jdt
~
a(V(t,x)),
for
some
a €
K;
if
further
(iii)
the
origin
of
the
x-space
belong
to
as;
(iv)
V(t,x)
= o
on
[t
0
,w)x(as
n
B,);
then
the
origin
is
unstable.
The
following
two
corollaries
of
Theorem
5.2.4
were
in
fact
established
long
before
Theorem
5.2.4
was
known.
Corollary
5.2.5
[Liapunov
1892]
If
there
exist
t
0
€
I,
212
E > o,
an
open
sets
c B.,
(with~
c
D),
and
a c
1
function
v:
[t
0
,oo)
xB. -+ R
such
that
on
[t
0
,oo)
xS:
(i)
0 <
V(t,x)
~
b(
llxll)
for
some b E
K;
(ii)
dV(t,x)jdt
~
a(llxll>
for
some a E
K;
if
further
(iii)
the
origin
of
the
x-space
belongs
to
as;
(iv)
V(t,x)
= 0
on
{t
0
,oo)x(as
n
B.);
then
the
origin
is
unstable.
Corollary
5.2.6
[Liapunov
1892]
If
in
Corollary
5.2.5,
(ii)
is
replaced
by
(ii')
V(t,x)
= c
V(t,x)
+
W(t,x)
on
[t
0
,oo)xS,
where
c > O,
and
W:
[t
0
,oo)xS -+ R
is
continuous
and
~
O,
then
the
origin
is
unstable.
Note
that,
if
the
differential
equation
is
autonomous
and
if
v
depends
on
x
only,
then
(i)
and
(ii)
in
Theorem
5.2.4
can
be
simplified
to:
(i)
V(x)
> 0
on
S;
and
(ii)
dV(x)jdt
> 0
on
s.
The
main
use
of
Corollary
5.2.6
is
to
help
prove
instability
by
considering
the
linear
approximation.
This
is
a
very
useful
way
to
look
at
many
applications.
Suppose,
dxjdt
=
f(t,x)
can
be
specified
as:
dxjdt
=
Ax
+
g(t,x)
(5.2-8)
where
A
is
an
nxn
real
matrix
and
Ax
+
g(t,x)
has
all
the
properties
required
from
f(t,x).
Then
we
have
the
following
theorem.
Theorem
5.2.7
[Liapunov
1892)
If
at
least
one
eigenvalue
of
A
has
strictly
positive
real
parts
and
if
llg(t,x)
ll!llxll -+ 0
as
x -+ 0
uniformly
for
t E
I,
then
the
origin
is
unstable
for
Eq.(5.2-8).
This
theorem
has
very
important
applications,
in
particular,
when
the
system
can
be
linearized.
There
are
many
practical
systems
which
satisfy
the
above
criteria
of
decomposing
f(t,x)
into
Ax+
g(t,x).
We
shall
encounter
this
theorem
later
in
Chapter
7.
As
an
example
for
immediate
demonstration,
we
shall
briefly
discuss
the
Watt's
governor.
It
is
a
well-known
device
and
it
is
sufficient
to
present
it
as
in
Fig.5.2.2.
If
we
disregard
the
friction,
the
equations
of
motion
are:
213