42
Introduction
1.4
Stability and Linearization
Suppose we are studying a physical system whose states
x
are described
by points in
R
n
. Assume that the dynamics of the system is governed by a
given evolution equation
dx
dt
=
f
(
x
)
.
Let
x
0
be a
stationary point
of the dynamics, i.e.,
f
(
x
0
) = 0. Imagine that
we perform an experiment on the system at time
t
= 0 and determine that
the initial state is indeed
x
0
. Are we justiFed in predicting that the system
will remain at
x
0
for all future time? The mathematical answer to this
question is yes, but unfortunately it is probably not the question we really
wished to ask. Experiments in real life seldom yield exact answers to our
idealized models, so in most cases we will have to ask whether the system
will remain
near
x
0
if it started
near
x
0
. The answer to the revised question
is not always yes, but even so, by examining the evolution equation at
hand more carefully, one can sometimes make predictions about the future
behavior of a system starting near
x
0
. A simple example will illustrate some
of the problems involved. Consider the following two di±erential equations
on the real line:
x
±
(
t
)=

x
(
t
)
(1.4.1)
and
x
±
(
t
x
(
t
)
.
(1.4.2)
The solutions are, respectively,
x
(
x
0
,t
x
0
e

t
(1.4.3)
and
x
(
x
0
x
0
e
+
t
.
(1.4.4)
Note that 0 is a stationary point of both equations. In the Frst case, for
all
x
0
∈
R
, we have lim
t
→∞
x
(
x
0
) = 0. The whole real line moves toward
the origin, and the prediction that, if
x
0
is near 0 then
x
(
x
0
) is near 0, is
justiFed. On the other hand, suppose we are observing a system whose state
x
is governed by equation (0.1.2). An experiment telling us that at time
t
= 0,
x
(0) is approximately zero will certainly not permit us to conclude
that
x
(
t
) stays near the origin for all time, since all points except 0 move
rapidly away from 0. ²urthermore, our experiment is unlikely to allow us
to make an accurate prediction about
x
(
t
) because if
x
(0)
<
0,
x
(
t
) moves
rapidly away from the origin toward
∞
, but if
x
(0)
>
0,
x
(
t
) moves toward
+
∞
. Thus, an observer watching such a system would expect sometimes