18.2 Center Manifolds Depending on Parameters 253
Thus, we see that (18.2.7) is very similar to (18.1.10).
Before considering a specific example we want to point out an important
fact. By considering ε as a new dependent variable, terms such as
xiεj, 1 ≤ i ≤ c, 1 ≤ j ≤ p,
or
yiεj, 1 ≤ i ≤ s, 1 ≤ j ≤ p,
become nonlinear terms. In this case, returning to a question asked at the
beginning of this section, the parts of the matrices A and B depending
on ε are now viewed as nonlinear terms and are included in the f and g
terms of (18.2.2), respectively. We remark that in applying center manifold
theory to a given system, it must first be transformed into the standard
form (either (18.1.1) or (18.2.2)).
Example 18.2.1 (The Lorenz Equations). Consider the Lorenz equations
˙x = σ(y − x),
˙y = ¯ρx + x − y − xz, (x, y, z) ∈ R3
, (18.2.8)
˙z = −βz + xy,
where σ and β are viewed as fixed positive constants and ¯ρ is a parameter (note:
in the standard version of the Lorenz equations it is traditional to put ¯ρ = ρ−1).
It should be clear that (x, y, z) = (0, 0, 0) is a fixed point of (18.2.9). Linearizing
(18.2.9) about this fixed point, we obtain the associated matrix


−σ σ 0
1 −1 0
0 0 −β

 . (18.2.9)
(Note: recall, ¯ρx is a nonlinear term.)
Since (18.2.9) is in block form, the eigenvalues are particularly easy to compute
and are given by
0, −σ − 1, −β, (18.2.10)
with eigenvectors 

1
1
0

 ,


σ
−1
0

 ,


0
0
1

 . (18.2.11)
Our goal is to determine the nature of the stability of (x, y, z) = (0, 0, 0) for ¯ρ
near zero. First, we must put (18.2.9) into the standard form (18.2.2). Using the
eigenbasis (18.2.11), we obtain the transformation


x
y
z

 =


1 σ 0
1 −1 0
0 0 1




u
v
w

 (18.2.12)
with inverse 

u
v
w

 =
1
1 + σ


1 σ 0
1 −1 0
0 0 1 + σ




x
y
z

 , (18.2.13)
254 18. Center Manifolds
which transforms (18.2.9) into


˙u
˙v
˙w

 =


0 0 0
0 −(1 + σ) 0
0 0 −β




u
v
w


+
1
1 + σ


σ¯ρ(u + σv) − σw(u + σv)
−¯ρ(u + σv) + w(u + σv)
(1 + σ)(u + σv)(u − v)

 ,
˙¯ρ = 0. (18.2.14)
Thus, from center manifold theory, the stability of (x, y, z) = (0, 0, 0) near ¯ρ = 0
can be determined by studying a one-parameter family of first-order ordinary
differential equations on a center manifold, which can be represented as a graph
over the u and ¯ρ variables, i.e.,
Wc
(0) = (u, v, w, ¯ρ) ∈ R4
| v = h1(u, ¯ρ), w = h2(u, ¯ρ),
hi(0, 0) = 0, Dhi(0, 0) = 0, i = 1, 2 (18.2.15)
for u and ¯ρ sufficiently small.
We now want to compute the center manifold and derive the vector field on
the center manifold. Using Theorem 18.1.4, we assume
h1(u, ¯ρ) = a1u2
+ a2u¯ρ + a3 ¯ρ2
+ · · · ,
h2(u, ¯ρ) = b1u2
+ b2u¯ρ + b3 ¯ρ2
+ · · · . (18.2.16)
Recall from (2.1.27) that the center manifold must satisfy
N h(x, ε) = Dxh(x, ε) [Ax + f(x, h(x, ε), ε)]
− Bh(x, ε) − g(x, h(x, ε), ε) = 0, (18.2.17)
where, in this example,
x ≡ u, y ≡ (v, w), ε ≡ ¯ρ, h = (h1, h2),
A = 0,
B =
−(1 + σ) 0
0 −β
, (18.2.18)
f(x, y, ε) =
1
1 + σ
[σ¯ρ(u + σv) − σw(u + σv)],
g(x, y, ε) =
1
1 + σ
−¯ρ(u + σv) + w(u + σv)
(1 + σ)(u + σv)(u − v)
.
Substituting (18.2.16) into (18.2.17) and using (18.2.19) gives the two components
of the equation for the center manifold.
(2a1u + a2 ¯ρ + · · ·)
σ
1 + σ
¯ρ(u + σh1) − h2(u + σh1)
+ (1 + σ)h1 +
¯ρ
1 + σ
(u + σh1) −
h2
1 + σ
(u + σh1) = 0,
18.2 Center Manifolds Depending on Parameters 255
(2b1u + b2 ¯ρ + · · ·)
σ
1 + σ
¯ρ(u + σh1) − h2(u + σh1)
+ βh2 − (u + σh1)(u − h1) = 0. (18.2.19)
Equating terms of like powers to zero gives
u2
: a1(1 + σ) = 0 ⇒ a1 = 0,
βb1 − 1 = 0 ⇒ b1 =
1
β
, (18.2.20)
u¯ρ : (1 + σ)a2 +
1
1 + σ
= 0 ⇒ a2 =
−1
(1 + σ)2
,
βb2 = 0 ⇒ b2 = 0.
Then, using (18.2.21) and (18.2.16), we obtain
h1(u, ¯ρ) = −
1
(1 + σ)2
u¯ρ + · · · ,
h2(u, ¯ρ) =
1
β
u2
+ · · · . (18.2.21)
Finally, substituting (18.2.21) into (18.2.14) we obtain the vector field reduced
to the center manifold
˙u =
σ
1 + σ
u ¯ρ −
1
β
u2
+ · · · ,
˙¯ρ = 0. (18.2.22)
FIGURE 18.2.1.
In Figure 18.2.1 we plot the fixed points of (18.2.22) neglecting higher order terms
such as O(¯ρ2
), O(u¯ρ2
), O(u3
), etc. It should be clear that u = 0 is always a fixed
point and is stable for ¯ρ < 0 and unstable for ¯ρ > 0. At the point of exchange of
stability (i.e., ¯ρ = 0) two new stable fixed points are created and are given by
¯ρ =
1
β
u2
. (18.2.23)
256 18. Center Manifolds
A simple calculation shows that these fixed points are stable. In Chapter 20 we
will see that this is an example of a pitchfork bifurcation.
Before leaving this example two comments are in order.
1. Figure 18.2.1 shows the advantage of introducing the parameter as a new
dependent variable. In a full neighborhood in parameter space new solutions
are “captured” on the center manifold. In Figure 18.2.1, for each fixed
¯ρ we have a flow in the u direction; this is represented by the vertical lines
with arrows.
2. We have not considered the effects of the higher order terms in (18.2.22)
on Figure 18.2.1. In Chapter 20 we will show that they do not qualitatively
change the figure (i.e., they do not create, destroy, or change the stability
of any of the fixed points) near the origin.
End of Example 18.2.1
18.3 The Inclusion of Linearly Unstable Directions
Suppose we consider the system
˙x = Ax + f(x, y, z),
˙y = By + g(x, y, z),
˙z = Cz + h(x, y, z), (x, y, z) ∈ Rc
× Rs
× Ru
, (18.3.1)
where
f(0, 0, 0) = 0, Df(0, 0, 0) = 0,
g(0, 0, 0) = 0, Dg(0, 0, 0) = 0,
h(0, 0, 0) = 0, Dh(0, 0, 0) = 0,
and f, g, and h are Cr
(r ≥ 2) in some neighborhood of the origin, A is a
c × c matrix having eigenvalues with zero real parts, B is an s × s matrix
having eigenvalues with negative real parts, and C is a u×u matrix having
eigenvalues with positive real parts.
In this case (x, y, z) = (0, 0, 0) is unstable due to the existence of a
u-dimensional unstable manifold. However, much of the center manifold
theory still applies, in particular Theorem 18.1.2 concerning existence, with
the center manifold being locally represented by
Wc
(0) = (x, y, z) ∈ Rc
× Rs
× Ru
| y = h1(x), z = h2(x),
hi(0) = 0, Dhi(0) = 0, i = 1, 2 (18.3.2)
for x sufficiently small. The vector field restricted to the center manifold is
given by
˙u = Au + f u, h1(u), h2(u) , u ∈ Rc
. (18.3.3)