Information Technology Reference
In-Depth Information
where
0
1
@f
1
@x
1
@f
1
@x
2
@f
1
@x
n
@
A
@f
2
@x
1
@f
2
@x
n
@f
n
@x
1
@f
2
@x
2
A Dr
x
f D
j
xDx
0
(2.35)
@f
n
@x
2
@f
n
@x
n
and
0
@
1
A
@f
1
@
u
1
@f
1
@
u
2
@f
1
@
u
n
@f
2
@
u
1
@f
2
@
u
n
@f
n
@
u
1
@f
2
@
u
2
B Dr
u
f D
j
xDx
0
(2.36)
@f
n
@
u
2
@f
n
@
u
n
The eigenvalues of matrix A define the local stability features of the system:
Example 1.
Assume the nonlinear system
d
2
x
dt
2
C 2x
d
dt
C 2x
2
4x D 0
(2.37)
By defining the state variables x
1
D x, x
2
Dx the system can be written in the
following form
x
1
D x
2
x
2
D2x
1
x
2
2x
1
C 4x
1
(2.38)
It holds that x D 0 if f.x/D 0 that is .x
;
1
;x
;
2
/ D .0;0/ and .x
;
1
;x
;
2
/ D .2;0/.
Round the first equilibrium .x
;
1
;x
;
2
/ D .0;0/ the system's dynamics is written as
01
40
or
x
1
x
2
01
40
x
1
x
2
A Dr
x
f D
D
(2.39)
The eigenvalues of the system are
1
D 2 and
2
D2. This means that the fixed
point .x
;
1
;x
;
2
/ D .0;0/ is an unstable one.
Next, the fixed point .x
;
1
;x
;
2
/ D .2;0/ is analyzed. The associated Jacobian
matrix is computed again. It holds that
01
4 4
A Dr
x
f D
(2.40)
The eigenvalues of the system are
1
D2 and
2
D2. Consequently, the fixed
point .x
;
1
;x
;
2
/ D .2;0/ is a stable one.
