1
Applied Mathematics 105b
February 2008 reprint/revision of February 1997 text
J. R. Rice
Notes on solutions of (linearized) ode systems near critical points
Let
{
x
} denote a column of
n
functions
x
1
(
t
),
x
2
(
t
), .
..,
x
n
(
t
)
such that
{
x
}
= [
x
1
(
t
),
x
2
(
t
),
...,
x
n
(
t
)]
T
(the
T
means transpose, i.e., interchange rows and columns, and here shifts the row to a column).
We suppose that the
{
x
}
satisfy the system of ode's of the form
{
˙
x
}
=
{
f
({
x
})
} ,
called an
autonomous
system of equations (because the functions {
f
} have no explicit
dependence on t). That is equivalent to writing
dx
1
/
dt
=
f
1
(
x
1
,
x
2
,...,
x
n
) ,
dx
2
/
dt
=
f
2
(
x
1
,
x
2
,...,
x
n
) ,
... ,
dx
n
/
dt
=
f
n
(
x
1
,
x
2
, .
.,
x
n
) .
We may think of
{
x
}
as a point in a n-dimensional space.
(When
n
= 2 , we generally write
{
x
}
= [
x
(
t
),
y
(
t
)]
T
and
{
f
} = [
f
(
x,y
),
g
(
x,y
)]
T
, so that
dx/dt
=
f
(
x,y
)
and
dy/dt
g
(
x,y
) .)
Note that the functions
{
f
({
x
})
} uniquely determine the direction of a local trajectory
through point
{
x
}
in the n-dimensional space unless all of the
{
f
}
happen to vanish
simultaneously.
For
n
= 2
that corresponds to a trajectory direction through
(
x,y
)
in the phase
plane being determined by
dy/dx
=
g
(
x,y
)/
f
(
x,y
) , unless both
f
and
g
vanish.
A
critical point
(or
singular point
) is such a point
{
x
cp
}
for which
{
f
({
x
cp
})
} = {
0
}.
By
doing a Taylor series expansion of the functions
{
f
({
x
})
}
about the critical point, neglecting
terms beyond the linear in
x
–
x
cp
in the expansion, and then shifting the origin of the space to
the critical point, i.e., renameing
x
–
x
cp
as
x
, we therefore obtain the linearized equations
{
˙
x
}
=
[
A
] {
x
}
near the critical point.