Civil Engineering Reference
In-Depth Information
The eigenvalues
of the Jacobian matrix:
λ
⎛
f
f
⎞
11
12
[3.21]
J
=
⎜
⎟
f
f
⎝
⎠
21
22
are the solutions to the equation
f
−
λ
f
11
12
=
0
f
f
−
λ
21
22
The above determinant enables us to write the
characteristic equation in the form
(
)
(
)
2
2
[3.22]
λ
−+
f
f
λ
+
f
f
−
f
f
=−+=
λ
pr
λ
0
11
22
11
22
12
21
where and . The solution to the first
equation in the system [3.20] around the critical point is
()
()
p
=
tr
J
r
=
det
J
Ce
λ
+
t
De
λ
−
t
x
1
(
t
)
=
+
where the eigenvalues are the roots of the characteristic
equation [3.22], i.e.
(
)
2
(
)
f
+±
f
f
+
f
−
4
f
f
−
f
f
11
22
11
22
11
22
12
21
λ
±
=
2
The behavior of the stability of the dynamical system
depends on the value of the discriminant
(Figure 3.18). For ,
the local topology is focal (spiral), either stable ( ) or
unstable ( ). This case corresponds to eigenvalues that
are complex conjugates, which suggests that the focal
trajectories either tend toward the critical point (stable,
2
p
2
(
)
(
)
δ =
−
4
r
=
f
11
+
f
22
−
4
f
11
f
22
−
f
12
f
21
δ <
0
p
<
0
>
0
p
)
<
0
p
or tend away from it (unstable,
). The eigenvalues are
p
>
0
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