Linearization
Goal
: Investigate the local behavior of a nonlinear system of diﬀerential equations near
its equilibrium points by linearizing the system.
Required tools
:
Matlab
routine
pplane
; eigenvalues and eigenvectors.
Discussion
In the last lab (Lab # 11) you classiﬁed equilibrium points in a linear system of
equations as sources, sinks, centers and saddles. (Recall, an equilibrium point is a
sink
if all solutions which begin suﬃciently close to it converge to it; it is a
source
if all
solutions suﬃciently close to it move away from it; it is a
center
if all solutions which
begin suﬃciently close to it “loop around” it, i.e. they return to their initial position
after a ﬁnite amount of time; the equilibrium point is a
saddle
if some solutions converge
to it and some move away from it.) You also saw that for a linear system, the eigenvalues
of the corresponding matrix determine what kind of equilibrium point the origin must
be. In this lab, we investigate how to determine the nature of the equilibrium points of
a
nonlinear
system.
Assignment
(1)
Consider ﬁrst the linear system
±
x
0
=

3
x
+(
√
2)
y
y
0
=(
√
2)
x

2
y
(
*
)
(a) Use
pplane
with window

x
≤
10 and

y
≤
10 to plot several orbits for the
system (
*
). What kind of equilibrium point does the origin (0
,
0) seem to be?
Print out your plot.
(b) In the
pplane
“
Options
” menu, select “
Erase all solutions
” and use
pplane
with
window

x
≤
0
.
1and

y
≤
0
.
1 to plot several orbits for (
*
). Print out this
plot. Notice that all of the orbits seem to be tangent to one particular line
at the origin.
(c) Let
A
be the 2
×
2 matrix for the system (
*
). As in Lab #11, use the com
mand
>>
[B,D]=eig(A)
to ﬁnd the eigenvalues and corresponding eigenvectors for
A
. Use this infor
mation to prove that the behavior you observed in (a) is correct. How does
the line noted in (b) relate to the eigenvectors of
A
?
(2)
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 Spring '08
 CHO
 matlab, Eigenvectors, Equations, Vectors, Fundamental physics concepts, Linear system, Equilibrium point, Nonlinear system

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