3.1 Stable, Unstable, and Center Subspaces 33
• For any c ∈ R, V is invariant with respect to cA.
• For any integer n > 1, V is invariant with respect to An
.
• Suppose A1 and A2 are linear maps on Rn
and V is invariant with
respect to both A1 and A2. Then V is invariant with respect to A1 +
A2. From this result it also follows that for any finite number of
linear maps Ai, i = 1, . . . , n, with V invariant under each, V is also
invariant under
n
i=1 Ai.
Using each of these facts one can easily conclude that V is invariant
under the linear map
Ln(t) ≡ id + At +
1
2
A2
t2
+ · · · +
1
n!
An
tn
=
n
i=0
1
i!
Ai
ti
,
for any n, where id is the n × n identity matrix (and 0! ≡ 1). Now using
the fact that V is closed, and that Ln(t) converges to eAt
uniformly, we
conclude that V is invariant with respect to eAt
.
3.1b Some Examples
We now illustrate these ideas with three examples where for simplicity and
easier visualization we will work in R3
.
Example 3.1.1. Suppose the three eigenvalues of A are real and distinct and
denoted by λ1, λ2 < 0, λ3 > 0. Then A has three linearly independent eigenvectors
e1, e2, and e3 corresponding to λ1, λ2, and λ3, respectively. If we form the
3 × 3 matrix T by taking as columns the eigenvectors e1, e2, and e3, which we
write as
T ≡




...
...
...
e1 e2 e3
...
...
...



 , (3.1.9)
then we have
Λ ≡


λ1 0 0
0 λ2 0
0 0 λ3

 = T−1
AT. (3.1.10)
Recall that the solution of (3.1.2) through y0 ∈ R3
at t = 0 is given by
y(t) = eAt
y0 = eT ΛT −1
t
y0. (3.1.11)
Using (3.1.4), it is easy to see that (3.1.11) is the same as
y(t) = TeΛt
T−1
y0
= T


eλ1t
0 0
0 eλ2t
0
0 0 eλ3t

 T−1
y0
34 3. Invariant Manifolds: Linear and Nonlinear Systems
=




...
...
...
e1eλ1t
e2eλ2t
e3eλ3t
...
...
...



 T−1
y0. (3.1.12)
Now we want to give a geometric interpretation to (3.1.12). Recall from (3.1.5)
that we have
Es
= span{e1, e2},
Eu
= span{e3}.
Invariance
Choose any point y0 ∈ R3
. Then T−1
is the transformation matrix which changes
the coordinates of y0 with respect to the standard basis on R3
(i.e., (1, 0, 0),
(0, 1, 0), (0, 0, 1)) into coordinates with respect to the basis e1, e2, and e3. Thus,
for y0 ∈ Es
, T−1
y0 has the form
T−1
y0 =


˜y01
˜y02
0

 , (3.1.13)
and, for y0 ∈ Eu
, T−1
y0 has the form
T−1
y0 =


0
0
˜y03

 . (3.1.14)
Therefore, by substituting (3.1.13) (resp., (3.1.14)) into (3.1.12), it is easy to see
that y0 ∈ Es
(resp., Eu
) implies eAt
y0 ∈ Es
(resp., Eu
). Thus, Es
and Eu
are
invariant manifolds.
Asymptotic Behavior
Using (3.1.13) and (3.1.12), we can see that, for any y0 ∈ Es
, we have eAt
y0 → 0
as t → +∞ and, for any y0 ∈ Eu
, we have eAt
y0 → 0 as t → −∞ (hence the
reason behind the names stable and unstable manifolds).
See Figure 3.1.1 for an illustration of the geometry of Es
and Eu
.
End of Example 3.1.1
Example 3.1.2. Suppose A has two complex conjugate eigenvalues ρ ± i ω,
ρ < 0, ω = 0 and one real eigenvalue λ > 0. Then A has three real generalized
eigenvectors e1, e2, and e3, which can be used as the columns of a matrix T in
order to transform A as follows
Λ ≡


ρ ω 0
−ω ρ 0
0 0 λ

 = T−1
AT. (3.1.15)
From Example 3.1.1 it is easy to see that in this example we have
y(t) = TeΛt
T−1
y0
= T


eρt
cos ωt eρt
sin ωt 0
−eρt
sin ωt eρt
cos ωt 0
0 0 eλt

 T−1
y0. (3.1.16)
3.1 Stable, Unstable, and Center Subspaces 35
FIGURE 3.1.1. The geometry of Es
and Eu
for Example 3.1.1.
Using the same arguments given in Example 3.1.1 it should be clear that Es
=
span{e1, e2} is an invariant manifold of solutions that decay exponentially to zero
as t → +∞, and Eu
= span{e3} is an invariant manifold of solutions that decay
exponentially to zero as t → −∞ (see Figure 3.1.2).
End of Example 3.1.2
Example 3.1.3. Suppose A has two real repeated eigenvalues, λ < 0, and a
third distinct eigenvalue γ > 0 such that there exist generalized eigenvectors e1,
e2, and e3 which can be used to form the columns of a matrix T so that A is
transformed as follows
Λ =


λ 1 0
0 λ 0
0 0 γ

 = T−1
AT. (3.1.17)
Following Examples 3.1.1 and 3.1.2, in this example the solution through the
point y0 ∈ R3
at t = 0 is given by
y(t) = TeΛt
T−1
y0
= T


eλt
teλt
0
0 eλt
0
0 0 eγt

 T−1
y0. (3.1.18)
Using the same arguments as in Example 3.1.1, it is easy to see that Es
=
span{e1, e2} is an invariant manifold of solutions that decay to y = 0 as t → +∞,
and Eu
= span{e3} is an invariant manifold of solutions that decay to y = 0 as
t → −∞ (see Figure 3.1.3).
End of Example 3.1.3
36 3. Invariant Manifolds: Linear and Nonlinear Systems
FIGURE 3.1.2. The geometry of Es
and Eu
for Example 3.1.2(for ω < 0).
FIGURE 3.1.3. The geometry of Es
and Eu
for Example 3.1.3
The reader should review enough linear algebra so that he or she can
justify each step in the arguments given in these examples. We remark
that we have not considered an example of a linear vector field having a
center subspace. The reader can construct his or her own examples from
Example 3.1.2 by setting ρ = 0 or from Example 3.1.3 by setting λ = 0; we
leave these as exercises and now turn to the nonlinear system.
3.8 Exercises 65
iii. For vector field b), discuss the cases λ < µ, λ = µ, λ > µ. What are
the qualitative and quantitative differences in the dynamics for these three
cases? Describe all zero- and one-dimensional invariant manifolds for this
vector field. Describe the nature of the trajectories at the origin. In particular,
which trajectories are tangent to either the x1 or x2 axis?
iv. In vector field c), describe how the trajectories depend on the relative
magnitudes of λ and ω. What happens when λ = 0? When ω = 0?
v. Describe the effect of linear perturbations on each of the vector fields.
vi. Describe the effect near the origin of nonlinear perturbations on each of
the vector fields. Can you say anything about the effects of nonlinear perturbations
on the dynamics outside of a neighborhood of the origin?
We remark that vi) is a difficult problem for the nonhyperbolic fixed points. We will
study this situation in great detail when we develop center manifold theory and bifurcation
theory.
23. Give a characterization of the stable, unstable, and center subspaces for linear maps in
terms of generalized eigenspaces along the same lines as we did for linear vector fields
according to the formulae (3.1.6), (3.1.7), and (3.1.8).
24. For the following linear vector fields find the general solution, and compute the stable,
unstable, and center subspaces and plot them in the phase space.
a)
˙x1
˙x2
=
1 2
3 2
x1
x2
b)


˙x1
˙x2
˙x3

 =


3 0 0
0 2 −5
0 1 −2




x1
x2
x3


c)


˙x1
˙x2
˙x3

 =


1 −3 3
3 −5 3
6 −6 4




x1
x2
x3


d)


˙x1
˙x2
˙x3

 =


−3 1 −1
−7 5 −1
−6 6 −2




x1
x2
x3


e)


˙x1
˙x2
˙x3

 =


1 0 0
1 2 0
1 0 −1




x1
x2
x3


f)


˙x1
˙x2
˙x3

 =


1 0 1
0 0 −2
0 1 0




x1
x2
x3


g)


˙x1
˙x2
˙x3

 =


0 0 15
1 0 −17
0 1 7




x1
x2
x3


h)


˙x1
˙x2
˙x3

 =


0 0 1
0 1 2
0 3 2




x1
x2
x3


25. Consider the following linear maps on R2
.
a)
x1
x2
→
λ 0
0 µ
x1
x2
,
|λ| < 1
|µ| > 1
.
b)
x1
x2
→
λ 0
0 µ
x1
x2
,
|λ| < 1
|µ| < 1
.
c)
x1
x2
→
λ −ω
ω λ
x1
x2
, ω > 0.
d)
x1
x2
→
1 0
0 λ
x1
x2
, |λ| < 1.
66 3. Invariant Manifolds: Linear and Nonlinear Systems
e)
x1
x2
→
1 λ
0 1
x1
x2
, λ > 0.
f)
x1
x2
→
1 0
0 1
x1
x2
.
i. For each map compute all the orbits and illustrate them graphically on
the phase plane. Describe the stable, unstable, and center manifolds of the
origin.
ii. For map a), discuss the cases λ, µ > 0; λ = 0, µ > 0; λ, µ < 0; and λ < 0,
µ > 0. What are the qualitative differences in the dynamics for these four
cases? Discuss how the orbits depend on the relative magnitudes of the
eigenvalues. Discuss the attracting nature of the unstable manifold of the
origin and its dependence on the relative magnitudes of the eigenvalues.
iii. For map b), discuss the cases λ, µ > 0; λ = 0, µ > 0; λ, µ < 0; and λ < 0,
µ > 0. What are the qualitative differences in the dynamics for these four
cases? Describe all zero- and one-dimensional invariant manifolds for this
map. Do all orbits lie on invariant manifolds?
iv. For map c), consider the cases λ2
+ ω2
< 1, λ2
+ ω2
> 1, and λ + iω = eiα
for α rational and α irrational. Describe the qualitative differences in the
dynamics for these four cases.
v. Describe the effect of linear perturbations on each of the maps.
vi. Describe the effect near the origin of nonlinear perturbations on each of the
maps. Can you say anything about the effects of nonlinear perturbations
on the dynamics outside of a neighborhood of the origin?
We remark that vi) is very difficult for nonhyperbolic fixed points (more so than the
analogous case for vector fields in the previous exercise) and will be treated in great
detail when we develop center manifold theory and bifurcation theory.
26. Consider the following vector fields.
a)
˙x = y,
˙y = −δy − µx,
(x, y) ∈ R
2
.
b)
˙x = y,
˙y = −δy − µx − x2
,
(x, y) ∈ R
2
.
c)
˙x = y,
˙y = −δy − µx − x3
,
(x, y) ∈ R
2
.
d)
˙x = −δx − µy + xy,
˙y = µx − δy + 1
2 (x2
− y2
),
(x, y) ∈ R
2
.
e)
˙x = −x + x3
,
˙y = x + y,
(x, y) ∈ R
2
.
f)
˙r = r(1 − r2
),
˙θ = cos 4θ,
(r, θ) ∈ R
+
× S
1
.
g)
˙r = r(δ + µr2
− r4
),
˙θ = 1 − r2
,
(r, θ) ∈ R
+
× S
1
.
h)
˙θ = v,
˙v = − sin θ − δv + µ,
(θ, v) ∈ S
1
× R.
i)
˙θ1 = ω1,
˙θ2 = ω2 + θn
1 , n ≥ 1,
(θ1, θ2) ∈ S
1
× S
1
.
j)
˙θ1 = θ2 − sin θ1,
˙θ2 = −θ2,
(θ1, θ2) ∈ S
1
× S
1
.
k)
˙θ1 = θ2
1,
˙θ2 = ω2,
(θ1, θ2) ∈ S
1
× S
1
.
Describe the nature of the stable and unstable manifolds of the fixed points by drawing
phase portraits. Can you determine anything about the global behavior of the
manifolds?
In a), b), c), d), g), and h) consider the cases δ < 0, δ = 0, δ > 0, µ < 0, µ = 0, and
µ > 0. In i) and k) consider ω1 > 0 and ω2 > 0.