Page 67
Invariant Manifolds
There are two basic motivations for invariant manifolds. The Frst comes
from the notion of separatrices that we have seen in our study of planar
systems, as in the Fgures. We can ask what is the higher dimensional gen
eralization of such separatrices. Invariant manifolds provides the answer.
The second comes from our study of of linear systems:
˙
x
=
Ax,
x
∈
R
n
.
Let
E
s
,
E
c
, and
E
u
be the (generalized) real eigenspaces of
A
associated
with eigenvalues of
A
lying on the open left half plane, the imaginary axes,
and the open right half plane, respectively. As we have seen in our study of
linear systems, each of these spaces is invariant under the ±ow of ˙
x
=
Ax
and represents, respectively, a stable, center, and unstable subspace. We
want to generalize these notions to the case of
nonlinear systems.
Thus,
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document68
Introduction
invariant manifolds will correspond, intuitively, to “nonlinear eigenspaces.”
Let us call a subset
S
⊂
R
n
a
k
manifold
if it can be
locally
represented
as the graph of a smooth function deFned on a
k
dimensional a±ne sub
space of
R
n
. As in the calculus of graphs,
k
manifolds have well deFned
tangent spaces at each point and these are independent of how the mani
folds are represented (or parametrized) as graphs. Although the notion of
a manifold is much more general, this will serve our purposes.
A
k
manifold
S
⊂
R
n
is said to be
invariant
under the ²ow of a vector
Feld
X
if for
x
∈
S
,
F
t
(
x
)
∈
S
for small
t>
0, where
F
t
(
x
) is the ²ow of
X
. One can show that this is equivalent to the condition that
X
is tangent
to
S
. One can thus say that
an invariant manifold is a union of (segments
of) integral curves of
X
.
While one can study invariant manifolds associated to general invariant
sets, such as periodic orbits, let us focus on Fxed points, say,
x
e
to begin—
these correspond to the origin for a linear system. There will be three
sorts of invariant manifolds, namely
stable manifolds
,
center mani
folds
, and
unstable manifolds
. In a neighborhood of
x
e
, the tangent
spaces to the stable, center, and unstable manifolds are provided by the
generalized eigenspaces
E
s
,
E
c
, and
E
u
of the linearization
A
=
DX
(
x
e
).
We are going to start with
hyperbolic points
; that is, points where
the linearization has no center subspace. Let the dimension of the stable
subspace be denoted
k
.
Theorem
(Local Invariant Manifold Theorem for Hyperbolic Points)
.
As
sume that
X
is a smooth vector Feld on
R
n
and that
x
e
is a hyperbolic
equilibrium point. There is a
k
 manifold
W
s
(
x
e
)
and a
n

k
manifold
W
u
(
x
e
)
each containing the point
x
e
such that the following hold:
i.
Each of
W
s
(
x
e
)
and
W
u
(
x
e
)
is locally invariant under
X
and con
tains
x
e
.
ii.
The tangent space to
W
s
(
x
e
)
at
x
e
is
E
s
and the tangent space to
W
u
(
x
e
)
at
x
e
is
E
u
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Marsden
 Mechanical Systems, Manifold, Dynamical systems, stable manifold, Center manifold, Center Manifold Theorem

Click to edit the document details