Download THE HOPF BIFURCATION AND ITS APPLICATIONS SECTION 2

THE HOPF BIFURCATION AND ITS APPLICATIONS
27
SECTION 2
THE CENTER MANIFOLD THEOREM
In this section we will start to carry out the program
outlined in Section 1 by proving the center manifold theorem.
The general invariant manifold theorem is given in HirschPugh-Shub [1].
Most of the essential ideas are also in
Kelley [1] and a treatment with additional references is contained in Hartman [1].
However, we shall follow a proof
given by Lanford [1] which is adapted to the case at hand,
and is direct and complete.
We thank Professor Lanford for
allowing us to reproduce his proof.
The key job of the center manifold theorem is to
enable one to reduce to a finite dimensional problem.
In the
case of the Hopf theorem, it enables a reduction to two dimensions without losing any information concerning stability.
The outline of how this is done was presented in Section 1
and the details are given in Sections 3 and 4.
In order to begin, the reader should recall some
results about basic spectral theory of bounded linear operators by consulting Section 2A.
The proofs of Theorems 1.3
THE HOPF BIFURCATION AND ITS APPLICATIONS
28
and 1.4 are also found there.
Statement and Proof of the Center Manifold Theorem
We are now ready for a proof of the center manifold
theorem.
It will be given in terms of an invariant manifold
~,
for a map
not necessarily a local diffeomorphism.
Later
we shall use it to get an invariant manifold theorem for flows.
Remarks on generalizations are given at the end of the proof.
(2.1)
Theorem.
Center Manifold Theorem.
Let
a mapping of a neighborhood of zero in a Banach space
Z.
~
We assume that
is
D~(O)
further assume that
D~(O)
the spectrum of
ck + l
, k > 1
~
be
Z
into
~(O)= O.
and that
has spectral radius
1
We
and that
splits into a part on the unit circle
and the remainder which is at a non-zero distance from the
unit circle.*
D~(O)
Y
Let
denote the generalized eigenspace of
belonging to the part of the spectrum on the unit circle;
assume that
Y
has dimension
d <
00.
V
Then there exists a neighborhood
submanifold
o and tangent to
a)
of
at
Y.
V
0,
of dimension
~.
If
and
d, passing through
x EM
and~(x)
E V,
(x) E M
(Local Attractivity):
n = 0, 1, 2, ••• ,
from
Z
in
such that
(Local Invariance):
then
b)
M
0
of
~n(x)
to
~n(x) E V
If
then, as
M
n
~
00,
for all
the distance
goes to zero.
We begin by reformulating (in a slightly more general
way) the theorem we want to prove.
We have a mapping
neighborhood of zero in a Banach space
Z
into
Z,
~
with
*This holds automatically if Z is finite dimensional or,
more generally, if D~(O) is compact.
of a
THE HOPF BIFURCATION AND ITS APPLICATIONS
=
~(o)
O.
We assume that the spectrum of
D~(O)
29
splits into
a part on the unit circle and the remainder, which is contained in a circle of radius strictly less than one, about the
origin.
The basic spectral theory discussed in Section 2A
guarantees the existence of a spectral projection
P
of
Z
belonging to the part of the spectrum on the unit circle with
the following properties:
i)
and
P
(I-P)Z
PZ
D~(O).
The spectrum of the restriction of
D~(O)
to
lies on the unit circle, and
iii)
to
so the subspaces
D~(O),
are mapped into themselves by
ii)
PZ
commutes with
(I-P)Z
We let
The spectral radius of the restriction of
is strictly less than one.
X denote
striction of
D (0)
to
D~(O)
(I-P)Z,
D~(O)
Y.
Then
~(x,y)
to
Z
X
=
Y
denote
and
X e Y
B
PZ,
A
denote the res-
denote the restriction of
and
(Ax+X*(x,y),
By + Y*(x,y»,
where
A
is bounded linear operator on
X with spectral
radius strictly less than one.
B
is a bounded operator on
Y with spectrum on
the unit circle.
(All we actually need is that
the spectral radius of B- 1 is no larger than one.)
k
X* is a c + l mapping of a neighborhood of the
origin in
X e Y
into
zero at the origin, i.e.
DX(O,O)
Y
Ck + l
is a
in
=
X
0,
X with a second-order
X(O,O)
=
0
and
and
mapping of a neighborhood of the origin
e Y into Y with a second-order zero at the
30
THE HOPF BIFURCATION AND ITS APPLICATIONS
origin.
~
We want to find an invariant manifold for
Y
to
at the origin.
mapping
into
u
X,
which is tangent
Such a manifold will be the graph of a
which maps a neighborhood of the origin in
with
=
u(O)
0
and
Du(O)
Y
= o.
In the version of the theorem we stated in 2.1, we
assumed that
Y
was finite-dimensional.
We can weaken this
assumption, but not eliminate it entirely.
(2.2)
Assumption.
¢ on
function
Y
1
which is
I Iyl
origin and zero for
k l
c +
There exists a
on a neighborhood of the
I > 1.
Perhaps surprisingly, this
assumption is actually rather restrictive.
if
Y
real-valued
is finite-dimensional or if
Y
It holds trivially
is a Hilbert space; for
a more detailed discussion of when it holds, see Bonic and
Frampton [1].
We can now state the precise theorem we are going to
prove.
(2.3)
Theorem.
Let the notation and assumptions be as
above.
Then there exist
{y E Y:
Ilyll
< d
e: > 0
into
X,
ck-mapping
and a
u*
from
with a second-order zero at zero,
such that
a)
IIYII
~
<d
c
X Ell Y, i.e. the graph of
in the sense that, i f
(xl'Yl)
II ylll
with
b)
r u* = ((x,y)
The manifold
<
e:
The manifold
in the sense that, i f
(~'Yn)
~n(x,y)
for all
n > 0,
Ilxll
then
<
e:
=
x
u*(y)
and
is invariant for
u*
~
and if
(u* (y) ,y)
Xl = u* (Yl) .
is locally attracting for
r u*
are such that
then
<
Ilyll
I
e:
Ilyll
II xnll
<
<
e:
e:
,
and if
IIYnl1
<
e:
~
THE HOPF BIFURCATION AND ITS APPLICATIONS
lim
n+ oo
I IXn
- u*(Yn)
II
31
o.
=
Proceeding with the proof, it will be convenient to
assume that
than
1.
IIA
II
< 1
and that
IIB-lil
is not much greater
This is not necessarily true but we can always make
it true by replacing the norms on
(See Lemma 2A.4).
X, Y
by equivalent norms.
We shall assume that we have made this
change of norm.
It is unfortunately a little awkward to exl
plicitly set down exactly how close to one
I IB- , I should
be taken.
We therefore carry out the proof as if
1 IB-li
I
were an adjustable parameter; in the course of the argument,
I IB- l , I.
we shall find a finite number of conditions on
In
principle, one should collect all these conditions and impose
them at the outset.
The theorem guarantees the existence of a function
u*
defined on what is perhaps a very small neighborhood of zero.
Rather than work with very small values of
x,y,
scale the system by introducing new variables
calling the new variables again
does not change
make
<
X*, Y*,
k + 1,
tiplying
A,B,
x
and
but, by taking
y).
s
we shall
xis,
y/s
(and
This scaling
very small, we can
together with their derivatives of order
as small as we like on the unit ball.
X*(x,y),
Y*(x,y)
by the function
Then by mul-
¢(y)
whose
existence is asserted in the assumption preceding the statement of the theorem, we can also assume that
are zero when
I Iyl I
> 1.
sup
II x 11::1
y unrestricted
x*(x,y), Y*(x,y)
Thus, if we introduce
sup
jl,j2
jl+j2::k+l
THE HOPF BIFURCATION AND ITS APPLICATIONS
32
we can make
A as small as we like by choosing
E
The only use we make of our technical assumption on
very small.
Y
is to
A
arrange things so that the supremum in the definition of
may be taken over all
y
and not just over a bounded set.
¢,
Once we have done the scaling and cutting off by
we can prove a global center manifold theorem.
That is, we
shall prove the following.
(2.4)
Lemma.
Keep the notation and assumptions of the
A is SUfficiently small (and if
center manifold theorem. If
liB-II
I
is close enough to one), there exists a function
defined and
k
u*,
Y,
times continuously differentiable on all of
with a second-order zero at the origin, such that
a)
invariant for
b)
lim
n+ oo
I Ix n
ru*
The manifold
"xii
If
As with
n
u*(y),y E Y}
{(x,y) Ix
is
in the strict sense.
0/
- u * (y
=
)
II
=
I IB-li
<
0
I,
1,
and
y
is arbitrary then
(where (xn'Yn)
=
we shall treat
0/
n
(x,y)).
A as an adjustable
parameter and impose the necessary restrictions on its size as
they appear.
It may be worth noting that
A depends on the
choice of norm; hence, one must first choose the norm to make
I /B- l , I close to one, then do the scaling and cutting off to
A small.
make
To simplify the task of the reader who wants.
to check that all the required conditions on
I IB- l "
can be satisfied simultaneously, we shall note these conditions
*
with a
as with (2.3)* on p. 34.
The strategy of proof is very simple.
manifold
of
u);
M of the form
we let
{x
= u(y)} (this stands for the graph
M denote the image of
some mild restrictions on
u,
We start with a
M under
0/.
With
we first show that the manifold
THE HOPF BIFURCATION AND ITS APPLICATIONS
~M
33
again has the form
{x = u(y)}
A
for a new function
u.
If we write ffu
for
u
we get a (non-
linear) mapping
u 1+'9'"u
from functions to functions.
and only if
u = ffu,
The manifold
M is invariant if
so we must find a fixed point of
do this by proving that
~
We
ff is a contraction on a suitable
A is small enough).
function space (assuming that
More explicitly, the proof will be divided into the
following steps:
I)
II)
Derive heuristi~ally a "formula" for
Show that the formula obtained in
Jv.
I) yields a
well-defined mapping of an appropriate function space
U
into
itself.
III)t Prove that
ff is a contraction on
has a unique fixed point
IV)
U
and hence
u*.
Prove that b) of Lemma (2.4) holds for
u*.
We begin by considering Step I).
I)
i)
y.
=
This means that
ii)
~(u(y),y),
we should proceed as follows
Solve the equation
y
for
u(y),
To construct
Let
By + Y*(u(y) ,y)
y
u(y)
is the
be the
Y-component of
of functions
~(u(y),y).
X-component of
Le.,
u(y) = Au(y) +X*(u(y),y).
II)
(2.1)
(2.2)
We shall somewhat arbitrarily choose the space
u
we want to consider to be
tone could use the implicit function theorem at this step.
For this approach, see Irwin [1].
THE HOPF BIFURCATION AND ITS APPLICATIONS
34
. {u: Y .... X I Dk+l u
U
j
0,1, ••• ,k+l,
continuous;
all
y;
U{O)
for
=
=
Du{O)
o}.
We must carry out two steps:
Prove that, for any given
i)
y
a unique solution
y E Y.
for each
that~,
Prove
ii)
u E U,
equation (1) has
And
defined by (2.2) is in
U.
To accomplish (i), we rewrite (2.1) as a fixed-point
problem:
It suffices, therefore, to prove that the mapping
_
Y 1+ B-1 Y -
is a contraction on
Y.
-)
B-1 Y * (u (-)
Y ,y
We do this by estimating its deriva-
tive:
I IDy[B-1Y-B-ly*(U(Y),y)] I I ::11B-lll
by the definitions of
A
and
l
+ D2 Y*(U(y) ,y) II :: 2AIIB- ll
U.
If we require
l
2AIIB- ll
< 1,
Y
is a function of
y,
By the inverse function theorem,
of
y.
Next we establish (ii).
and
< 1
for all
= 0,
D.%.(O)
First take
j = 0:
-%.(0)
IliVu(y)11
y.
Note
y
is a
Ck + l
function
By what we have just proved,
Thus to show ~ E U,
I I DjiVu(y) II
for each
*
depending also on the function
u.
5'u E c k + l •
(2.3)
y
equation (2.1) has a unique solution
that
IID1Y*(u(y),y)Du(y)
y,
what we must check is that
j = 0,1,2, ..• ,k+l
(2. 4)
(2.5)
= O.
~ IIAII·llu(Y)11 + Ilx*(u(Y),Y)11
< IIAII
+ A,
so if we require
IIAII + A < 1,
then
I Wu{y)
II
< 1
for all
y.
(2.6)
*
THE HOPF BIFURCATION AND ITS APPLICATIONS
To estimate
D9U
we must first estimate
35
Dy(y).
By
differentiating (2.1), we get
(2.7)
yU: Y
where
+
Y
is defined by
=
yU(y)
Y*(u(y) ,y).
By a computation we have already done,
I IDYu(y)!!
Now
B + DY
u
~ 2A
B[I+B-1DY u )
(2.3)*), B + DY u
for all
and since
y.
2AI IB-11!
<
1
(by
is invertible and
The quantity on the right-hand side of this inequality will
play an important role in our estimates, so we give it a name:
(2.8)
1IB- l , I
Note that, by first making
by making
A
y
small, we can make
very close to one and then
as close to one as we like.
We have just shown that
II Dy (y) II
< y
for all
Differentiating the expression (2.2) for
D5U(y)
=
y.
YU(y)
(2.9)
yields
[A DU(y) + DXu(y») Dy(y); }
and
(2.10)
(xu (y) = X*(u(y) ,y».
Thus
11D9U(y) II
<
(IIAII + 2A) .y,
(2.11)
36
THE HOPF BIFURCATION AND ITS APPLICATIONS
so if we require
(IIAII
+ 2A) Y
(2.12)
< 1,
we get
11D%(y)
II
< 1
for all
y.
We shall carry the estimates just one step further.
Differentiating (2.7) yields
By a straightforward computation,
so
Now, by differentiating the formula (2.10) for
D~,
we get
2
D :§ilu(y)
so
If we require
(2.13)
we have
IID2~(y)11 <1
for all
y.
At this point it should be plausible by imposing a sequence of
stronger and stronger conditions on
I I Djy(y) II
< 1
for all
y,
Y,A,
that we can arrange
j = 3,4, •.. ,k+1.
*
*
THE HOPF BIFURCATION AND ITS APPLICATIONS
37
The verification that this is in fact possible is left to the
reader.
~u
To check (2.5), i.e.
u = 0,
Du
yeo)
o
0)
IL%. (0)
o
D%
(assuming
we note that
since
0
is a solution of
AU(O) + X(u(O) ,0) = 0
~u(O)
0,
By + Y(u(y) ,y)
0
and
[A Du(O) + D1X(0,0)Du(0) + D2X(0,0)] ·Dy(O)
[A· 0 + 0 + 0]· Dy (0) = O.
This completes step II). Now we turn to III)
III)
We show that
~
is a contraction and apply the
contraction mapping principle.
What we actually do is slightly
more complicated.
i)
norm.
Since
We show that
U
~
is a contraction in the supremum
is not complete in the supremum norm, the con~
traction mapping principle does not imply that
point in
U,
but it does imply that
the completion of
ii)
U
has a fixed
~ has a fixed point in
with respect to the supremum norm.
We show that the completion of
U
with respect
to the supremum norm is contained in the set of functions
from
to
y
X
th
k
with Lipschitz-continuous
and with a second-order zero at the origin.
point
u*
~
of
derivatives
Thus, the fixed
has the differentiability asserted in the
theorem.
We proceed by proving
i)
I I u l -u 2 I 10
=
Consider
i).
u l ' u 2 E U,
supl lu l (y)-u 2 (y) I I·
y
Let
and let
Yl(y), Y2(y)
the solution of
y
u
1,2.
denote
38
THE HOPF BIFURCATION AND ITS APPLICATIONS
We shall estimate successively
11:i\-Y211 0' and
Subtracting the defining equations for
Yl'Y2'
II9"u 1 ~211 O.
we get
so that
(2.14)
Since
I I Du 1 1 10 2 1,
we can write
(2.15)
Inserting (2.15) in (2.14) and rearranging, yields
1
(l-n·IIB- 11) IIY1-Y211
I 1Y1 -Y 2 1I 0
i. e.
2. >.·IIB-lll·llu2-ulII0'
~
>.. yo
I I u 2 - u 1 1I •
I
J
(2.16)
Now insert estimates (2.15) and (2.16) in
to get
If we now require
a
3t
=
II All (l+y>.) + >. (l+2y>.)
< 1,
will be a contraction in the supremum norm.
(2.17)*
39
THE HOPF BIFURCATION ANO ITS APPLICATIONS
ii)
The assertions we want all follow directly from
the following general result.
(2.5)
Lemma.
a Banach space
Y
that, for all
Let
(un)
be a sequence of functions on
with values on a Banach space
nand
Assume
y E Y,
j
O,1,2, ..• ,k,
is Lipschitz continuous with Lipschitz
and that each
constant one.
(un(y))
X.
Assume also that for each
y,
the sequence
converges weakly (i.e., in the weak topology on
to a unit vector
a)
u
u(y).
X)
Then
k th derivative
has a Lipschitz continuous
with Lipschitz constant one.
b)
y
and
ojun(y)
converges weakly to
oju(y)*
for all
j = 1, 2 , ••• , k •
If
X, Yare finite dimensional, all the Banach space
technicalities in the statement of the proposition disappear,
and the proposition becomes a straightforward consequence of
the Arzela-Ascoli Theorem.
We postpone the proof for a moment,
and instead turn to step IV).
IV)
Ilxll
< 1
We shall prove the following:
and let
y E Y
be arbitrary.
Let
Let
(xl'Yl)
* This statement may require some interpretation.
n,y,ojun(y)
x E X with
'I'(x,y) .
For each
j
y
is a bounded symmetric j-linear map from
to X. What we are asserting is that, for each Y'Yl' ..• 'Yj'
the sequence (ojun(y) (Yl' •.• 'Yj)) of elements of X converges in the weak topology on X to oju(y) (Yl' ... 'Yj).
THE HOPF BIFURCATION AND ITS APPLICATIONS
40
Then
II xIII
~ 1
and
~ CY.·llx-u*(y) II,
Ilxl-u*(Yl) II
where
CY.
is as defined in (2.17).
[ I x n -u* (Y n ) II
< cy'nll x-u* (Y)
(2.18)
By induction,
II
as
0
-+
n
-+
00,
as asserted.
To prove
I Ixll
I < 1,
we first write
Xl = Ax + X(x,y),
/lxlll
~
IIAlj·llxll
+ It ~
so that
"All
+ It
< 1
by
(2.6)
To prove (2.18), we essentially have to repeat the estimates made in proving that
j7
is a contraction.
Let
Yl
the solution of
On the other hand, by the definition of
Y
l
=
Y
l
we have
By + Y(x,y).
Subtracting these equations and proceeding exactly as in the
derivation of (2.16), we get
Next, we write
xl = Ax + X(x,y).
Subtracting and making the same estimates as before, we get
be
41
THE HOPF BIFURCATION AND ITS APPLICATIONS
as desired.
This completes step IV).
Let us finish the argument by supplying the details for
Lemma (2.5).
Proof of Lemma (2.5)
We shall give the argument only for
ization to arbitrary
k
k = 1; the general-
is a straightforward induction argu-
ment.
Yl'Y2 E Y
We start by choosing
and
¢ E X*
and con-
sider the sequence of real-valued functions of a real variable
From the assumptions we have made about the sequence
(un)'
it follows that
lim ~n(t) = ¢(u(Yl+tY2))
n->-QO
for all
t,
~n(t)
that
I~~ (t)1
<
=
~(t)
is differentiable, that
II ¢ 1/ •II y 1//
for all
n, t
and that
for all
n, t l , t .
2
By this last inequality and the Arzela-
Ascoli Theorem, there exists a subsequence
verges uniformly on every bounded interval.
rily denote the limit of this subsequence by
hence, passing to the limit
j
->- 00,
we get
~'
n j (t)
which con-
We shall temporaX(t).
We have
THE HOPF BIFURCATION AND ITS APPLICATIONS
42
1jJ(t) =1jJ(O) +
which implies that
1jJ (t)
f:
X(T) dT,
is continuously differentiable and
that
1jJ
I
(t)
X(t) •
To see that
lim1jJ~(t)
= 1jJ'(t)
nT OO
(i.e., that it is not necessary to pass to a subsequence), we
note that the argument we have just given shows that any subsequence of
(1jJ' (t»
n
has a subsequence converging to
1jJ
I
(t) ;
this implies that the original sequence must converge to this
limit.
Since
we conclude that the sequence
converges in the weak topology on
shall denote by
only suggestive.
DU(Yl) (Y2);
we have
to a limit, which we
this notation is at this point
By passage to a limit from the correspond-
is a bounded linear mapping of norm
for each
X**
< 1
from
We denote this linear operator by
yto
X **
THE HOPF BIFURCATION AND ITS APPLICATIONS
i.e., the mapping
to
~
y
Du(y)
43
is Lipschitz continuous from
Y
L(Y,X ** ).
The next step is to prove that
this equation together with the norm-continuity of
will imply that
u
is
(Frechet)-differentiable.
Du(y)
y~
The integral
in (2.19) may be understood as a vector-valued Riemann integral.
By the first part of our argument,
for all
¢
E X**
and taking Riemann integrals commutes with
continuous linear mappings, so that
for all
¢
*
EX.
Therefore (2.19) is proved.
The situation is now as follows:
if we regard
u
as a mapping into
X**
We have shown that,
which contains
then it is Frechet differentiable with derivative
other hand, we know that
want to conclude that
x.
X
u
actually takes values in
t+O
DU(Yl) (Y2)
and
belongs to
u(Yl+ tY 2)-u(Yl)
t
the difference quotients on the right all belong to
is norm closed in
X
But
~~l~
X
On the
it is differentiable as a mapping into
This is equivalent to proving that
for all
Du.
X,
X**
the proof is complete. CJ
X,
and
44
THE HOPF BIFURCATION AND ITS APPLICATIONS
(2.6)
Remarks on the Center Manifold Theorem
It may be noted that we seem to have lost some
1.
differentiability in passing from
that
~
is
ck + l
to
and only concluded that
fact, however, the
k th
~
u*
u* ,
since we assumed
u*
ck .
is
In
we obtain has a Lipschitz continuous
derivative, and our argument works just as well if we
~
only assume that
k th
has a Lipschitz continuous
deriva-
tive, so in this class of maps, no loss of differentiability
occurs.
k th
Moreover, if we make the weaker assumption that the
derivative of
~
is uniformly continuous on some neigh-
borhood of zero, we can show that the same is true of
(Of course, if
X
and
u*.
Yare finite dimensional, continuity
on a neighborhood of zero implies uniform continuity on a
neighborhood of zero, but this is no longer true if
X
or
Y
is infinite dimensional).
2.
As
C. Pugh has pointed out, if
~
is infinite-
ly differentiable, the center manifold cannot, in general, be
taken to be infinitely differentiable.
that, if
~
It is also not true
is analytic there is an analytic center manifold.
We shall give a counterexample in the context of equilibrium
points of differential equations rather than fixed points of
maps; cf. Theorem
2.7 below.
This example, due to Lanford,
also shows that the center manifold is not unique;
cf. Exercise 2.8.
Consider the system of equations:
~
0,
= -
.where
zero.
h
(2.20 )
is analytic near zero and has a second-order zero at
We claim that, if
h
is not analytic in the whole com-
THE HOPF BIFURCATION AND ITS APPLICATIONS
p1ex plane, there is no function
neighborhood of
(0,0)
u(Y1'Y2)'
45
analytic in a
and vanishing to second order at
(0,0),
such that the manifold
is locally invariant under the flow induced by the differential
equation near
(0,0).
To see this, we assume that we have an
invariant manifold with
Straightforward computation shows that the expansion coefficients
c,
,
are uniquely determined by the requirement of
J 1 ,J2
invariance and that
(j1+ j 2) :
(j1):
h j1 + j2 ,
where
h(Y1)
If the series for
series for
h
u(o'Y2)
= L
j.:. 2
j
h 'Y1
J
has a finite radius of convergence, the
diverges for all non-zero
Y2'
The system of differential equations has nevertheless
many infinitely differentiable center manifolds.
one, let
h(Y1)
agreeing with
To construct
be a bounded infinitely differentiable function
h
on a neighborhood of zero.
Then the manifold
defined by
(2.21)
is easily verified to be globally invariant for the system
THE HOPF BIFURCATION AND ITS APPLICATIONS
46
0,
dx
dt
(2.22)
and hence locally invariant at zero for the original system.
(To make the expression for
sketch its derivation.
volve
x
u
less mysterious, we
The equations for
Yl'Y2
and are trivial to solve explicitly.
do not inA function
u
defining an invariant manifold for the modified system (2.22)
must satisfy
for all
t, Yl' Y2.
The formula (2.21) for
u
is obtained
by solving this ordinary differential equation with a suitable
boundary condition at
3.
~
t
=
_00.)
Often one wishes to replace the fixed point
by an invariant manifold
on a normal bundle of
9.
V.
V
0
of
and make spectral hypotheses
We shall need to do this in section
This general case follows the same procedure; details are
found in Hirsch-Pugh-Shub [lJ.
The Center Manifold Theorem for Flows
The center manifold theorem for maps can be used to
prove a center manifold theorem for flows.
time
t
We work with the
maps of the flow rather than with the vector fields
themselves because, in preparation for the Navier Stokes equations, we want to allow the vector field generating the flow
to be only densely defined, but since we can often prove that
the time
t-maps are
00
C
this is a reasonable hypothesis for
many partial differential equations (see
tails) •
Section 8A for de-
47
THE HOPF BIFURCATION AND ITS APPLICATIONS
(2.7)
Let
z
Theorem.
Center Manifold Theorem for Flows.
and let
Ft
o < t <
zero for
Ft(X)
cO
be a
ck + l
is
C
0
semi flow defined in a neighborhood of
Assume
T.
norm * away from
00
be a Banach space admitting a
jointly in
Ft(O)
t
and
and that for
0,
=
x.
t > O.
Assume that the
spectrum of the linear semi group DF (0): Z -+ Z is of the
t
tal
form e t (01U0 2 ) where
lies
on
the unit circle (Le.
e
ta
2 lies inside the
°1 lies on the imaginary axis) and e
unit circle a nonzero distance from it, for
is in the left half plane.
Let
Y
t> 0; i.e.
02
be the generalized eigen-
space corresponding to the part of the spectrum on the unit
circle.
a
o
Assume
dim Y
d <
00.
Then there exists a neighborhood
V
of
submanifold
d
passing through
and tangent to
M CV
Y
at
of dimension
0
0
in
Z
and
such that
(a)
If
x EM, t > 0
(b)
If
t > 0
and
then
Ft(x) E V,
Ft(X) EM
V
for all
n
0,1,2, .•• ,
and
F~(X)
then
remains defined and in
F~(X)
-+
M as
n
-+
This way of formulating the result is the most convenient for it applies to ordinary as well as to partial differential equations, the reason is that we do not need to worry
about "unboundedness" of the generator of the flow.
Instead
we have used a smoothness assumption on the flow.
The center manifold theorem for maps, Theorem 2.1,
applies to
V
and
M
F , t > O. However, we are claiming that
t
can be chosen independent of t. The basic reason
e~ch
48
THE HOPF BIFURCATION AND ITS APPLICATIONS
for this is that the maps~t}
commute:
Fs
F t -- F t+s-
0
F
0
F ' where defined. However, this is somewhat overs
t
simplified. In the proof of the center manifold theorem we
would require the
F
to remain globally commuting after they
t
$.
have been cut off by the function
That is, we need to
ensure that in the course of proving lemma 2.4,
A can be
chosen small (independent of
are globally
t)
and the
Ft'S
defined and commute.
The way to ensure this is to first cut off the
Z
outside a ball
B
in such a way that the
turbed in a small ball about
tity outside of
of
F
t
0
00
function
C
and is
0
outside
where
T =
it is easy to see that
Z
G
t
which coincides with
f
B.
remains a
Ft , 0
semiflow.
t
and
and are the iden-
x
which is one on a neighThen defining
(2.23)
extends to a global semiflow t on
~
t
~ T
on a neighborhood of
B.
Moreover,
Gt
(For this to be true we required
the smoothness of the norm on
smoothness in
T,
Z
and that for
jointly * )
t > 0
.
F
t
Now we can rescale and chop off simultaneously the
outside
F
t
in
are not dis-
It f(F (x))ds,
o
s
zero, and which is the identity outside
Ck + l
-<
t
t
This may be achieved by joint continuity
B.
and use of a
borhood of
0, 0 < t
F
F
B as in the above proof.
has
G
t
Since this does not affect
on a small neighborhood of zero, we get our result.
*In
linear semi group theory this corresponds to analyticity
of the semigroup; it holds for the heat equation for instance.
For the Navier Stokes eqations, see Sections 8,9.
t see Renz [1] for further details.
THE HOPF BIFURCATION AND ITS APPLICATIONS
Exercise.
(2.8)
for flows).
d(x,y)
dt
=
(nonuniqueness of the center manifold
X (x,y) = (-x,y 2 ).
Let
X(x,y)
49
Solve the equation
and draw the integral curves.
Show that the flow of
Theorem 2.7 with
X
satisfies the conditions of
the y-axis.
y
center manifold for the flow.
Show that the
y-axis is a
Show that each integral curve
in the lower half plane goes toward the origin as
that the curve becomes parallel to the y-axis as
t
t
~
and
->-
-00.
Show
that the curve which is the union of the positive y-axis with
any integral curve in the lower half plane is a center manifold
for the flow of
X.
(see Kelley [1], p. 149).
Exercise.
(2.9)
(Assumes a knowledge of linear semi-
group theory).
Consider a Hilbert space
space) and let
A
Let
be a
K: H
H
->-
H
(or a "smooth" Banach
be the generator of an analytic semigroup.
Ck + l
mapping and consider the evolution
equations
dx
dt
(a)
on
H
Ax + K(x),
x(O)
= X
(2.24)
o
Show that these define a local semiflow
c k +l
which is
in (t,x)
Duhamel integral equation
for
t > O.
x(t) = eAtx
o
+
Ft(x)
(Hint: Solve the
t-
f
0 e (
t-s ) AK(x(s))ds
by iteration or the implicit function theorem on a suitable
space of maps (references: Segal [1], Marsden [1], Robbin [1]).
(b)
of
i~variant
Assume
K(O)
=
0, DK(O)
=
O.
Show the existence
manifolds for (2.24) under suitable spectral
hypotheses on
A
by using Theorem 2.7.
50
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 2A
SOME SPECTRAL THEORY
In this section we recall quickly some relevant results
in spectral theory.
Schwartz [1,2].
For details, see Rudin [1] or Dunford-
Then we go on and use these to prove Theorems
1. 3 and 1.4.
Let
space
E
T: E
and let
{A E ~[T - AI
Then
I ITI
a(T)
I·
+
Let
E
be a bounded linear operator on a Banach
a(T)
denote its spectrum,
a(T) =
is not invertible on the complexification of
is non-empty, is compact, and for
r(T)
r(T) = SUp{IAI
A E a(T),IAI
El
<
denote its spectral radius defined by
IA E a(T)}.
r(t)
is also given by the spectral
radius formula:
r(T)
limit I ITnl
I
lin
•
n+oo
(The proof is analogous to the formula for the radius of convergence of a power series and can be supplied without difficulty; cf. Rudin [1, p.355].)
The following two lemmas are also not difficult and are
THE HOPF BIFURCATION AND ITS APPLICATIONS
51
proven in the references given:
(2A.l)
Lemma.
tion and define
(2A.2)
Lemma.
d(01,02) > 0,
E
l
and
o (T )
i
E2
o .•
1.
f(T)
Let
= La
L a n zn
f(z)
Tn
then
On'
Suppose
otT)
0 (f (T» = f (0 (T» •
E
i
E
=
El
e
E2
0 1 U 02
=
then there are unique
such that
where
T-invariant subspaces
and if
=
Ti
TIE.'
then
1.
is called the generalized eigenspace of
Let
Lemma 2A.2 is done as follows:
curve with
be an entire func-
o
in its interior and
O
2
0i'
be a closed
in its exterior,
then
A is not al-
Note that the eigenspace of an eigenvalue
ways the same as the generalized eigenspace of
A.
In the
finite dimensional case, the generalized eigenspace of
A is
the subspace corresponding to all the Jordan blocks containing
A in the Jordan canonical form.
(2A. 3)
Lemma.
Let T, 0 1 , and O 2 be as in Lemma
°1 °2
2A.2 and assume that d(e , e ) > O. Then for the operator
tT
the generalized eigenspace of e i
e tT
is Ei •
Proof.
Write, according to Lemma 2A.2,
E
Thus,
From this the result follows easily.
(2A.4)
Lemma.
Let
T: E
+
E
[]
be a bounded, linear
THE HOPF BIFURCATION AND ITS APPLICATIONS
52
operator on a Banach space
E.
Let
r
be any number greater
than
r(T),
the spectral radius of
I
on
equivalent to the original norm such that ITI
I
E
We know that
Proof.
II Tnll
sup
Therefore,
n~a
we see that
(~~~
r sup
n~a
I
E
r
n
I
:s
and let
Let
r > cr(A)
I = sUb
A: E
+
Ie
of
I
tAl
I
E
i.e.
n~a
r
II Tn(x) II
n
r
n
be a bounded operator
A E
cr(A) , Re
on
E
A<
r).
such that for
Then
t > a,
< e
rt.
Note that (see Lemma 2A.l) e
etA;
sup
I}/n.
II Tn+l(x) II
n~
(i.e. if
there is an equivalent norm
Proof.
=
r.
0
rl xl .
Lemma.
I xl
<
Ilxll.2lxl.2
I T(x)
Hence
I l'7nl
r(T) = lim
I f we define
< "'.
r
II Tn+l(X) II
n+l
r
Then there is a norm
is given by
is a norm and that
I I Tnn " ) I I x I I .
(2A.5)
on
r(t)
T.
rt
e rt > lim I I entAl I lin.
is
>
spectral radius
Set
n+'"
Ixl = sup
n;::a
II entA(x) II
e
rnt
t~a
and proceed as in Lemma 2A.4.
There is an analogous lemma if
CJ
r
< cr (A) ,
giving
e
rt
With this machinery we now turn to Theorems 1.3 and 1.4 of
Section 1:
(1.3)
operator.
Let
Theorem.
Let
T: E
cr(T) C {zlRe z < a}
+
E
be a bounded linear
(resp.
cr(T) ~ {zlRe z > a}),
then the origin is an attracting (resp. repelling) fixed point
for the flow
¢t = e tT
of
T.
'!'HE POPF BIFURCA,TION AND ITS APPLICATIONS
Proof.
{Z
This is immediate from Lemma 2A.5 for if
IRe z < O},
is compact.
53
there is an r < 0 with 0 (T) < r,
tA
Thus
le
I.:5 e rt .... 0 as t .... +00. D
0(T)~
as
0 (T)
Next we prove the first part of Theorem 1.4 from Section 1.
(1.4)
Theorem.
Banach manifold
P
Let
X
cl
be a
and be such that
o (dX (p 0)) C {z E <c IRe z < O},
vector field on a
=
X(PO)
then
O.
If
is asymptotically
stable.
Proof.
Po
=
O.
We can assume that P is a Banach space E
As above, renorm
I letA I I < e - d ,
where
A
E
and find
€ > 0
and that
such that
dX (0) •
=
From the local existence theory of ordinary differen-
. * , there is a
tial equat10ns
r-ball about
for which the
0
time of existence is uniform if the initial condition
lies in this ball.
such that
where
o
Let
R(x)
Find
X
X(x) - DX(O) ·x.
=
Ilx II
2 r 2 implies
IIR(x)11
<a.llxll
a. = €/2.
Let
show that if
remains in
B
be the open
X
E B,
o
Band
....
r /2 ball about O. We shall
2
then the integral curve starting at X
0
exponentially as
t
.... +00.
o
This will
prove the result.
Let
Suppose
x (t)
x(t)
be an integral curve of
remains in
B
for
0
<t
X
< T.
starting at
Then noting
that
*See
for instance, Hale [1], Hartman [1], Lang [1], etc.
x •
o
THE HOPF BIFURCATION AND ITS APPLICATIONS
54
x
x(t)
= x
I:
I:
+
o
+
o
x (x (s)) ds
A(x(s)) + R(x(s)) ds
gives (the Duhamel formula; Exercise 2.9),
x(t) = e tAx o + It
0 e (t-s)A R(x(s)) ds
I:
and so
Ilx(t)11
Thus, letting
~e-tExo
f (t) = e
tE
Ct
Ijx(t)
~xO
f(t)
+
+
Ct
e-(t-s)Ellx(s)llds.
II,
I:
f(s) ds
and so
(This elementary inequality is usually called Gronwall's inequality cf. Hartman [1].)
Thus
Hence
x(t) E B,
extended in
x(t)
+
0
t
as
0 < t
< T
so
x(t)
may be indefinitely
and the above estimate holds.
t
+
+00.
It shows
~
The instability part of Theorem 1.4 requires use of
the invariant manifold theorems, splitting the spectrum into
two parts in the left and right half planes.
The above analysis
shows that on the space corresponding to the spectrum in the
right half plane,
(2A.6)
PO
is repelling, so is unstable.
Remarks.
Theorem 1.3 is also true by the same
THE HOPF BIFURCATION AND ITS APPLICATIONS
55
proof if T is an unbounded operator, provided we know
o (e tT )
etO(T) which usually holds for decent T,
(e.g.:
if the spectrum is discrete) but nee,d not always be true (See
Hille-Phillips [1]).
This remark is important for applications
to partial differential equations.
require
which is unrealistic for partial
R(x) = 0(1 Ixl I)
differential equations.
Assume
F
in
t
However, the following holds:
O(DFt(O»
=
etO(DX(O»
and we have a
cO flow
cl
with derivative strongly continuous
(See Section 8A),
then the conclusion of 1.4 is true.
and each
t
Theorem 1.4's proof would
F
t
is
in the notation above,
This can be proved as follows:
we have, by Taylor's theorem:
where
R(t,x)
as long as
is the remainder.
x
remains in a small neighborhood of
-2Et
< e
and hence
because
some
E > 0
R(t,x)
=
J:
Now we will have
and
x
in some neighborhood of
{DFt(sx),x - DFt(O) ·x} ds.
x
and
00
+
0
Exercise.
(2A.7)
F: E
+
E
a
exponentially as
cl
map with
inside the unit circle.
U
of
0
Let
such that if
is a stable point of
E
t
+
for
remember
remains close to
0
~
be a Banach space and
F(O) = 0
and the spectrum of
DF(O)
Show that there is a neighborhood
x EU, F(n) (x)
F.
-Et
< e
Therefore, arguing as
in Theorem 1.4, we can conclude that
Ft(x)
0 '"
this is
0;
+
0
as
n
+
00;
Le.
0
56
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 2B
THE POINCAR~ MAP
We begin by recalling the definition of the Poincare
map.
In doing this, one has to prove that the mapping exists
and is differentiable.
cO
of
flows
Ft(X)
In fact, one can do this in the context
such that for each
t, F
t
ck ,
is
as
was the case for the center manifold theorem for flows, but
here with the additional assumption that
t
t > 0.
as well for
Ft(X)
is smooth in
Again, this is the appropriate hypo-
thesis needed so that the results will be applicable to partial
differential equations.
However, let us stick with the ordinary
differential equation case where
vector field
X
F
t
is the flow of a
at first.
First of all we recall that a closed orbit
flow
on a manifold
Yet)
X
L >
y
O.
ck
such that
The least such
M
of a
is a non-constant integral curve
y(t+L)
L
y
Yet)
for all
is the period of
y.
t ER
and some
The image of
is clearly diffeomorphic to a circle.
(2B.l)
Definition.
Let
y
be a closed orbit,
let
THE HOPF BIFURCATION AND ITS APPLICATIONS
m E Y,
say
m
Y(O)
=
and let
S
57
be a local transversal sec-
tion; i.e. a submanifold of codimension one transverse to
,
Y(Le.
Y (0)
is not tangent to
S).
'21 eM
Let
x IR
be the
domain (assumed open) on which the flow is defined.
A Poincare map of
(PM 1)
(PM 2)
there is a function
p(x)
(PM 3)
if
=
o
->-
W
l
where:
are open neighborhoods of
k diffeomorphism;
is a C
P
for all
P: W
S
WO'W l e
and
is a mapping
Y
0: W
o
->-
IR
m E S,
such that
x E WO' (X,T-O(X)) E'21 and
F(X,T-O(X)); and finally,
t E (O,T-O(X)), then
~
F(x,t)
W
o
(see Figure 2B.l).
(2B.2)
Theorem (Existence and uniqueness of Poincare
maps).
(i)
If
X
is a closed orbit of
is a
ck
vector field on
M,
and
Y
then there exists a Poincare map of
X,
Y.
(in
->-
Wl
a local transversal section
S
at
m E
and
pI
are locally conjugate.
(ii)
S'
at
P: W
o
If
m ' E Y),
then
P
is a Poincare map
That is, there are open neighborhoods
m' E S I,
W e Wo
2
and a
n
Wl '
k
C
and
~
I:
0
pI
Y
also
(in
,
of
W
of m E S, W
2
2
,
H: W ->- W '
such that
2
2
diffeomorphism
I
W2 e Wo and the diagram
,
1
0- (W,)
;
j
.W
W2----1.8....'--»,
commutes.
y)
of
n
0W,
THE HOPF BIFURCATION AND ITS APPLICATIONS
58
P(x)= F(x,t - 8{x»
Figure 2B.l
Here is the idea of the proof of existence of
further details see Abraham-Marsden [1).
trarily.
By continuity,
borhood
U
o
assumption,
of
m.
of
at
x
U
2
t-differentiable at
=
m
P
P
tive of
FT-O(x) (x) E S.
This is
P(x),
will be as differentiable as
at
m
FT
arbi-
of
m.
t = 0
By
and is
and hence also in a neighborhood
Therefore, there is a unique number
such that
tion
S
is
S
(for
is a homeomorphism of a neigh-
to another neighborhood
FT+t(X)
transverse to
of
m
FT
Choose
P
F
o(x)
near zero
and by construcis.
The derivative
is seen to be just the projection of the derivaon
a diffeomorphism,
T S
m
P
(this is done below).
will be as well.
particular, this means
Ft
t > 0
FT
is
CJ
For partial differential equations,
a semi-flow i.e. defined for
Hence if
F
is often just
t
(See Section 8A). In
is not generally a diffeomorphism.
THE HOPF BIFURCATION AND ITS APPLICATIONS
(For instance if
59
is the flow of the heat equation on L ,
2
t
is not surjective). For the Poincare mapping this means
F
F
t
that we will have a
for
t > 0),
morphism.
but
P
P
(if
Ft(X)
is differentiable in
t, x
is just a map, not necessarily a diffeo-
This is one of the technical reasons why it is im-
portant to have the center manifold theorem for maps and not
just diffeomorphisms.
As we outlined in Section 1, there is a Hopf theorem
for diffeomorphisms and this is to be eventually applied to
the Poincare map
circle for
P
P
to deduce the existence of an invariant
and hence an invariant torus for
Ft.
For partial differential equations, this seems like a
dilemma since
P
need not be a diffeomorphism.
can be overcome by a trick:
However, this
first apply the center manifold
theorem to reduce everything to finite dimensions--this does
not require diffeomorphisms; then, as proved in Section BA,
(see BA.9) in finite dimensions
F
and hence P will autot
matically become (local) diffeomorphisms. Thus the dilemma is
only apparent.
Let us now prove the fundamental results concerning the
derivative of
P
of
Ft.
E,
and we can let
so we can relate the spectrum of
P
to that
It suffices to do the computation in a Banach space
m = 0,
the origin of
We begin by calculating
(2B.3)
Lemma.
Banach space.
Let
dP(O)
Let
in terms of
F : E + E be a
t
be on a closed orbit
0
E.
Ft.
Cl
flow on a
y
with period
dF
f O.
Let
ated by
V
T
E = F
e
V
t
~ (0) It=o = V.
and let
and
F
Let
V
be the subspace gener-
be a complementary subspace, so that
1
2
(Ft(X,y), Ft(X,y)). Let P: F + F
Ft{x,y)
60
THE HOPF BIFURCATION AND ITS APPLICATIONS
be the Poincare map associated with
y
at the point
O.
Then
2
dP (0) = d1F 'I (0) .
Proof.
Let
'I(X)
the above notation).
be the time at
Then by definition of
P(x)
so
aF
dP(x)
Letting
P(x)
1
(x)
in
P,
1
F 'I (x) (x, 0)
1
(X,O)d'I(X) + d1F'I(x) (X,O).
at
(x,O) = 0,
('I-O (x)
we get
a F1
1
(O)d'I(O) + d1F (0) .
'I
2
aF
)
(O,V)
by construc(0), at'I (0)
at
dP (0) =
1
aF
'I
However d t (0) = (a F
a F1
'I
tion,so
a t (0) = o. Thus,
-IT-
~
dF
(2B.4)
Lemma.
Proof.
~ (0,0)
(0,0)
ds
= dF
'I
So,
dF of
'I s
ds
Is=O
dF t
= dt
(0,0)
It='I
v
(O,O)ls=o
(d1F'I(0,0)'0 + d 2 F 'I (0,0)V)
(0,0) (O,V)
d F (0,0)V
2 'I
( 2B• 5 )
0
d F (0,0)V = V.
2 'I
dF 'I + S
Is=O
1
d1F'I(0).
dP (0)
V.
Lemma.
0
CJ (dF'I (0 , 0) )
CJ (P (0» U {l}.
(This is
true of the point spectrum, too).
Proof.
*
where
Let
A
The matrix of
dF (0,0)
'I
is
indicates some unspecified matrix entry.
E~.
Then
A
E CJ(dF'I(O,O»
iff
dF'I(O,O) -
AI
is not
THE HOPF BIFURCATION AND ITS APPLICATIONS
I - l o r is not onto.
But
(dF T (0, 0) - AI) (~) = (dP (
(dP (0) 0: - Ao:,
Clearly,
°)0:,
1 Ea(dFT(O,O».
(0:,6)
we can choose
6
Since
The former implies
A 'I 1,
for any
0:,
There-
1'1 A Ea(dP(O}}.
On the other hand, let
is not onto, then clearly neither is
dFT(O,O} - AI.
0.
A Ea(dFT(O,O».
so that the second component is onto.
A Eo(dP(O)}.
dP(O) - AI
and
such that the expression
Assume the latter.
=
A '11
is zero or the map is not onto.
fore,
+ 6) + (- A0: , - A6 )
(2B. 1)
Assume
A E a(dP(O».
If,
* (0:)
* (0:) + 6 (1 - A».
Then either, there exists
(2B.l)
61
Suppose there exists
Then choosing
6
=
*(o:)
'-1'
0:
such that
we see that
dP(O}o: - Ao:
A E o:(dF (0,0».
0
T
A
Consult Abraham-Marsden [1] Abraham-Robbin [1] or Hartman
[1, Section IV. 6, IX.IO] for additional details on this and
and the associated Floquet theory.
One of the most basic uses of the Poincare map is in
the proof of the following.
(2B.6)
with period
Theorem.
T.
Let
Let
m E y
be a closed orbit for
y
and suppose
dFT(m)
inside the unit circle except for one point
circle.
Then
y
Proof.
1
F
t
has spectrum
on the unit
is an attracting (stable) closed orbit.
By Demma 2B.5, the condition on the spectrum
means that the spectrum of the derivative of the Poincare map
P
at
m
is inside the unit circle.
an attracting fixed pojnt for
struction of
P
that
y
P.
Hence from 2A.7
m
is
It fo110ws from the con-
is attracting for
Ft.
o
62
THE HOPF BIFURCATION AND ITS APPLICATIONS
(2B.7)
Exercise.
(a)
Give the details in the last step of the above
(b)
If, in 2B.6,
proof.
P
has an attracting invariant
circle, give the details of the proof that this yields an
attracting invariant 2 torus for the flow.