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In-Depth Information
For x
2
Dx<0one obtains the state-space description
x
1
D x
2
(2.25)
k
m
x
1
c
m
x
2
C
k
g
x
2
D
2.2
Computation of Isoclines
An autonomous second order system is described by two differential equations of
the form
x
1
D f
1
.x
1
;x
2
/
x
2
D f
2
.x
1
;x
2
/
(2.26)
The method of the isoclines consists of computing the slope (ratio) between f
2
and
f
1
for every point of the trajectory of the state vector .x
1
;x
2
/.
f
2
.x
1
;x
2
/
f
1
.x
1
;x
2
/
s.x/ D
(2.27)
The case s.x/ D c describes a curve in the x
1
x
2
plane along which the trajectories
x
1
D f
1
.x
1
;x
2
/ and x
2
D f
2
.x
1
;x
2
/ have a constant slope.
The curve s.x/ D c is drawn in the x
1
x
2
plane and along this curve one also
draws small linear segments of length c. The curve s.x/ D c is known as isocline.
The direction of these small linear segments is according to the sign of the ratio
f
2
.x
1
;x
2
/=f
1
.x
1
;x
2
/.
Example 1.
The following simplified pendulum equation is considered
x
1
D x
2
x
2
Dsin.x
1
/
(2.28)
The slope s.x/ is given by the relation
f
2
.x
1
;x
2
/
sin.x
2
/
x
2
s.x/ D
f
1
.x
1
;x
2
/
)s.x/ D
(2.29)
Setting s.x/ D c it holds that the isoclines are given by the relation
1
c
sin.x
1
/
(2.30)
x
2
D
For different values of c one has the following isoclines diagram depicted in Fig.
2.4
.
