Chemistry Reference
In-Depth Information
8
<
:
2a
1
d
1
LN
.N
1/x
2
;
x
1
.t
C
1/
D
.1
a
1
/x
1
.t/
C
T
j
D
.1/
W
2a
2
d
2
LN
x
1
.t/
.N
2/x
2
.t/
;
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
and the equilibrium outputs are the solutions of the algebraic system
2d
1
LN
.N
1/x
2
;
x
1
D
2d
2
LN
x
1
.t/
.N
2/x
2
.t/
;
x
2
D
.1/
. The local asymptotic sta-
provided that these solutions are inside the region
D
.1/
bility of any equilibrium point inside region
D
is determined by the study of the
eigenvalues of the Jacobian matrix
0
1
2a
1
d
1
.N
1/
.LN
.N
1/x
2
/
2
1
a
1
@
A
J
.1/
D
:
2a
2
d
2
.LN
x
1
.N
2/x
2
/
2
1
a
2
C
2a
2
d
2
.N
2/
.LN
x
1
.N
2/x
2
/
2
computed at the equilibrium. However, boundary equilibria can also exist, located
in regions
.i /
, i
D
2;3;4. For example, in the region
.2/
, where the map assumes
D
D
the form
8
<
2a
1
d
1
LN
.N
1/x
2
;
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
a
2
L;
x
1
.t
C
1/
D
.1
a
1
/x
1
.t/
C
T
j
D
.2/
W
:
we can have a boundary equilibrium with coordinates E
D
.x
1
;x
2
/
D
.2d
1
=L;L/
2 D
.2/
, provide
d that x
2
<
NL
2
2d
1
.N
NL
2
1
2d
2
and x
2
>
2
x
1
C
2/L
. This holds if
p
2d
1
<L<
p
d
1
C
d
2
. It is clear that if the equilibrium E exists, that is if the
above inequalities are satisfied, then it is locally asymptotically stable, because
inside the region
1/L
N
.N
.2/
the Jacobian matrix is triangular with eigenvalues .1
a
1
/
and .1
a
2
/. Similar arguments apply to the region
D
.3/
, where the map assumes
D
the form
x
1
.t
C
1/
D
.1
a
1
/x
1
.t/
C
a
1
L;
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
a
2
L;
and a boundary equilibrium of coordinates E
0
D
x
0
1
; x
0
2
D
.L;L/
2 D
T
j
D
.3/
W
.3/
exists
x
0
2
>
NL
2
2d
1
NL
2
2d
2
x
0
2
>
1
2
x
0
1
provided that
and
C
2/L
, which imply that
.N
1/L
N
.N
L< min
p
2d
1
;
p
2d
2
. Whenever these inequalities are satisfied, then the bound-
ary point .L;L/ is a locally asymptotically stable equilibrium, since the Jacobian
Search WWH ::
Custom Search