3.2 Stable, Unstable, and Center Manifolds for Fixed Points 37
3.2 Stable, Unstable, and Center Manifolds for
Fixed Points of Nonlinear, Autonomous Vector
Fields
Recall that our original motivation for studying the linear system
˙y = Ay, y ∈ Rn
, (3.2.1)
where A = Df(¯x), was to obtain information about the nature of solutions
near the fixed point x = ¯x of the nonlinear equation
˙x = f(x), x ∈ Rn
. (3.2.2)
The stable, unstable, and center manifold theorem provides an answer to
this question; let us first transform (3.2.2) to a more convenient form.
We first transform the fixed point x = ¯x of (3.2.2) to the origin via the
translation y = x − ¯x. In this case (3.2.2) becomes
˙y = f(¯x + y), y ∈ Rn
. (3.2.3)
Taylor expanding f(¯x + y) about x = ¯x gives
˙y = Df(¯x)y + R(y), y ∈ Rn
, (3.2.4)
where R(y) = O(|y|2
) and we have used f(¯x) = 0. From elementary linear
algebra (see Hirsch and Smale [1974]) we can find a linear transformation
T which transforms the linear equation (3.2.1) into block diagonal form


˙u
˙v
˙w

 =


As 0 0
0 Au 0
0 0 Ac




u
v
w

 , (3.2.5)
where T−1
y ≡ (u, v, w) ∈ Rs
× Ru
× Rc
, s + u + c = n, As is an s × s
matrix having eigenvalues with negative real part, Au is an u × u matrix
having eigenvalues with positive real part, and Ac is an c × c matrix
having eigenvalues with zero real part (note: we point out the (hopefully)
obvious fact that the “0” in (3.2.5) are not scalar zero’s but rather the
appropriately sized block consisting of all zero’s. This notation will be used
throughout the book). Using this same linear transformation to transform
the coordinates of the nonlinear vector field (3.2.4) gives the equation
˙u = Asu + Rs(u, v, w),
˙v = Auv + Ru(u, v, w), (3.2.6)
˙w = Acw + Rc(u, v, w),
where Rs(u, v, w), Ru(u, v, w), and Rc(u, v, w) are the first s, u, and c
components, respectively, of the vector T−1
R(Ty
).
38 3. Invariant Manifolds: Linear and Nonlinear Systems
Now consider the linear vector field (3.2.5). From our previous discussion
(3.2.5) has an s-dimensional invariant stable manifold, a u-dimensional invariant
unstable manifold, and a c-dimensional invariant center manifold all
intersecting in the origin. The following theorem shows how this structure
changes when the nonlinear vector field (3.2.6) is considered.
Theorem 3.2.1 (Local Stable, Unstable, and Center Manifolds of
Fixed Points) Suppose (3.2.6) is Cr
, r ≥ 2. Then the fixed point
(u, v, w) = 0 of (3.2.6) possesses a Cr
s-dimensional local, invariant stable
manifold, Ws
loc(0), a Cr
u-dimensional local, invariant unstable manifold,
Wu
loc(0), and a Cr
c-dimensional local, invariant center manifold, Wc
loc(0),
all intersecting at (u, v, w) = 0. These manifolds are all tangent to the respective
invariant subspaces of the linear vector field (3.2.5) at the origin
and, hence, are locally representable as graphs. In particular, we have
Ws
loc(0) = (u, v, w) ∈ Rs
× Ru
× Rc
| v = hs
v(u), w = hs
w(u);
Dhs
v(0) = 0, Dhs
w(0) = 0; |u| sufficiently small
Wu
loc(0) = (u, v, w) ∈ Rs
× Ru
× Rc
| u = hu
u(v), w = hu
w(v);
Dhu
u(0) = 0, Dhu
w(0) = 0; |v| sufficiently small
Wc
loc(0) = (u, v, w) ∈ Rs
× Ru
× Rc
| u = hc
u(w), v = hc
v(w);
Dhc
u(0) = 0, Dhc
v(0) = 0; |w| sufficiently small
where hs
v(u), hs
w(u), hu
u(v), hu
w(v), hc
u(w), and hc
v(w) are Cr
functions.
Moreover, trajectories in Ws
loc(0) and Wu
loc(0) have the same asymptotic
properties as trajectories in Es
and Eu
, respectively. Namely, trajectories
of (3.2.6) with initial conditions in Ws
loc(0) (resp., Wu
loc(0)) approach the
origin at an exponential rate asymptotically as t → +∞ (resp., t → −∞).
Proof: See Fenichel [1971], Hirsch, Pugh, and Shub [1977], or Wiggins [1994]
for details as well as for some history and further references on invariant
manifolds. ⊓⊔
Some remarks on this important theorem are now in order.
Remark 1. First some terminology. Very often one hears the terms “stable
manifold,” “unstable manifold,” or “center manifold” used alone; however,
alone they are not sufficient to describe the dynamical situation. Notice
that Theorem 3.2.1 is entitled stable, unstable, and center manifolds of
fixed points. The phrase “of fixed points” is the key: one must say the
stable, unstable, or center manifold of something in order to make sense.
The “somethings” studied thus far have been fixed points; however, more
3.2 Stable, Unstable, and Center Manifolds for Fixed Points 39
general invariant sets also have stable, unstable, and center manifolds. See
Wiggins [1994] for a discussion.
Remark 2. The conditions Dhs
v(0) = 0, Dhs
w(0) = 0, etc., reflect that the
nonlinear manifolds are tangent to the associated linear manifolds at the
origin.
Remark 3. In the statement of the theorem the term local, invariant stable,
unstable, or center manifold is used. This deserves further explanation.
“Local” refers to the fact that the manifold is only defined in the neighborhood
of the fixed point as a graph. Consequently, these manifolds have
a boundary. They are therefore only locally invariant in the sense that trajectories
that start on them may leave the local manifold, but only through
crossing the boundary. Invariance is still manifested by the vector field
being tangent to the manifolds, which we discuss further below.
Remark 4. Suppose the fixed point is hyperbolic, i.e., Ec
= ∅. In this case
an interpretation of the theorem is that trajectories of the nonlinear vector
field in a sufficiently small neighborhood of the origin behave the same as
trajectories of the associated linear vector field.
Remark 5. In general, the behavior of trajectories in in Wc
loc(0) cannot be
inferred from the behavior of trajectories in Ec
.
Remark 6. Uniqueness of Stable, Unstable, and Center Manifolds. Typically
the existence of these invariant manifolds are proved through a contraction
mapping argument, where the invariant manifold turns out to be
the unique fixed point of an appropriately constructed contraction map.
From this construction the stable and unstable manifolds are unique. The
center manifold is a bit more delicate. In that case, because of the nonhyperbolicity,
a “cut-off” function is typically used in the construction of the
appropriate contraction map. In this case the center manifold does depend
upon the cut-off function. However, it can be shown that the center manifold
is unique to all orders of its Taylor expansion. That is, center manifolds
only differ by exponentially small functions of the distance from the fixed
point. See Wan [1977], Sijbrand [1985] and Wiggins [1994].
3.2a Invariance of the Graph of a Function:
Tangency of the Vector Field to the Graph
Suppose one has a general surface, or manifold and one wants to check if it
is invariant with respect to the dynamics generated by a vector field. How
can this be done?
Suppose the vector field is of the form
˙x = f(x, y),
˙y = g(x, y), (x, y) ∈ Rn
× Rm
.
Suppose that the surface in the phase space is represented by the graph of
40 3. Invariant Manifolds: Linear and Nonlinear Systems
a function
y = h(x),
This surface is invariant if the vector field is tangent to the surface. This
tangency condition is expressed as follows
Dh(x) ˙x = ˙y,
or,
Dh(x)f(x, h(x)) = g(x, h(x)). (3.2.7)
Of course, one must take care that all the functions taking part in these
expressions have common domains, and that the appropriate derivatives
exist. It is also very important to appreciate the role that specific coordinate
representations played in deriving this expression.
3.3 Maps
An identical theory can be developed for maps. We summarize the details
below. Consider a Cr
diffeomorphism
x → g(x), x ∈ Rn
. (3.3.1)
Suppose (3.3.1) has a fixed point at x = ¯x and we want to know the nature
of orbits near this fixed point. Then it is natural to consider the associated
linear map
y → Ay, y ∈ Rn
, (3.3.2)
where A = Dg(¯x). The linear map (3.3.2) has invariant manifolds given by
Es
= span{e1, · · · , es},
Eu
= span{es+1, · · · , es+u},
Ec
= span{es+u+1, · · · , es+u+c},
where s + u + c = n and e1, · · · , es are the (generalized) eigenvectors of
A corresponding to the eigenvalues of A having modulus less than one,
es+1, · · · , es+u are the (generalized) eigenvectors of A corresponding to the
eigenvalues of A having modulus greater than one, and es+u+1, · · · , es+u+c
are the (generalized) eigenvectors of A corresponding to the eigenvalues of
A having modulus equal to one. The reader should find it easy to prove this
by putting A in Jordan canonical form and noting that the orbit of the
linear map (3.3.2) through the point y0 ∈ Rn
is given by
{· · · , A−n
y0, · · · , A−1
y0, y0, Ay0, · · · , An
y0, · · ·}. (3.3.3)
Now we address the question of how this structure goes over to the nonlinear
map (3.3.1). In the case of maps Theorem 3.2.1 holds identically.