Chemistry Reference
In-Depth Information
The assumption that
g
.k/
.k/
can be extended to an open set containing
D
can be
replaced by the following. For the linear segment between
x
and any
x
2 D
there
are finitely many values 0
D
t
0
<t
1
<
<t
K.
x
/
D
1 such that
(a) Subsegments Œ
x
C
t
l
.
x
x
/;
x
C
t
lC1
.
x
x
/
D
.k
l
/
for all l
(b) For each such subsegment there are sequences
f
u
k
g
and
f
v
k
g
such that
u
k
!
x
C
t
l
.
x
x
/,
v
k
!
x
C
t
lC1
.
x
x
/ and the entire linear segment Œ
u
k
;
v
k
is
.k
l
/
.
in the interior of
D
Since there are infinitely many matrix norms, and the condition of the theorem
might hold with one matrix norm and not with others, the above conditions are
difficult to check in practical applications.
For example, in the cases of matrices
0:8 0
!
;
0:8 0:8
!
0:51 0:51
!
;
A
1
D
A
2
D
and
A
3
D
0:8 0
00
0:51 0
p
1:28
'
1:13 > 1;
k
A
1
k
1
D
0:8 < 1;
k
A
1
k
1
D
1:6 > 1;
k
A
1
k
2
D
p
1:28
'
1:13 > 1
k
A
2
k
1
D
1:6 > 1;
k
A
2
k
1
D
0:8 < 1;
k
A
2
k
2
D
and
k
A
3
k
1
D
1:02 > 1;
k
A
3
k
1
D
1:02 > 1;
s
3
C
p
5
2
k
A
3
k
2
D
.0:51/
'
0:825 < 1:
That is, only one of the most popular matrix norms is below one, the other two
norms are greater than one. It is well-known that if all eigenvalues of a matrix are
inside the unit circle, then there is a matrix norm such that the norm of this matrix
is below one, (see for example, Ortega and Rheinholdt (1970)). Therefore we can
reformulate Theorem B.1 as follows:
Theorem B.4.
Let
x 2 D
be an equilibrium and assume that
J
.
x
/
exists in an open
neighborhood of
x
, and is continuous at
x
. Assume furthermore that all eigenvalues
of
J
.
x
/
are inside the unit circle. Then
x
is locally asymptotically stable.
Unfortunately this eigenvalue criterion cannot be extended to prove global asymp-
totic stability. That is, the assumption that for all
x
the eigenvalues of
J
.
x
/
are inside the unit circle does not necessarily imply the
x
is globally asymptotically
stable. In fact Cima et al. (1997, 1999) present counterexamples for n
D
2 and n
3.
Instability of equilibria cannot be proved by showing that a particular matrix
norm of the Jacobian at
x
is larger than one, since there is the possibility that another
norm of the Jacobian is less than one. However it is well known that if at least one
eigenvalue of
J
.
x
/ is outside the unit circle, then
x
is unstable. For an elementary
proof see Li and Szidarovszky (1999
a
). If for all eigenvalues
i
of
J
.
x
/,
j
i
j
1
and at least one eigenvalue is located on the unit circle, then no conclusion can be
2 D
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