Civil Engineering Reference
In-Depth Information
An alternative way of analyzing the spatiotemporal
behavior of the system [3.18] is to eliminate the time factor
t
and consider the relation
(
)
f
XX
,
,
dX
X
1
1
2
[3.19]
1
X
==
1
(
)
dX
f
X
X
2
2
2
1
2
(
)
of the trajectories in the phase plane
instead of
X
1
,
X
2
determining the behaviors over time
()
and
()
. The
X
1
t
X
2
t
slopes
are clearly defined everywhere in the plane
, except at the so-called “critical” points where
dX
1
dX
2
(
)
X
1
,
X
2
simultaneously. Suppose that the critical points
XX
==
0
1
2
are located, respectively, at and . The
system is first expressed in relation to the critical points
and , and the new coordinate system is defined by
, . The behavior of the dynamical
system around new critical points
XX
==
0
X
10
X
20
1
2
X
10
X
20
x
1
=
X
1
−
X
10
x
2
=
X
2
−
X
20
is expressed in
x
10
=
x
20
=
0
a first-order Taylor series by
∂
f
∂
f
x
=
1
x
+
1
x
1
1
2
∂
x
∂
x
1
2
[3.20]
∂
f
∂
f
x
=
2
x
+
2
x
2
1
2
∂
x
∂
x
1
2
We now introduce . We determine the time-
derivative of the first equation in [3.20] and substitute it into
the second equation. Thus, we obtain
f
ij
= ∂
f
i
∂
x
j
(
)
x
=+ =+
fx
fx
fx
f
fx
+
f x
1
11
1
12
2
11
1
12
21
1
22
2
We eliminate
by using the first relation in [3.20].
x
2
Finally, we obtain
(
)
(
)
x
=+
f
f
xf
+
f
−
f
f
x
1
11
22
1
12
21
11
22
1
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