Hopf bifurcation and exchange of stability in diffusive media
T
HOMASB
RAND, M
ARKUSK
UNZE
, G
UIDOS
CHNEIDER
, T
HORSTENS
EELBACH
Mathematisches Institut, Universit¨at Bayreuth, D - 95440 Bayreuth, Germany
FB6 – Mathematik, Universit¨at Essen, D - 45117 Essen, Germany
Mathematisches Institut I, Universit¨at Karlsruhe, D - 76128 Karlsruhe, Germany
February 6, 2003
Abstract
We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a pair of imaginary eigenvalues crosses the imaginary axis. For a reaction-diffusion-convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a Hopf bifurcation and the nonlinear stability of the bifurcating time-periodic solutions, again with respect to spatially localized pertur- bations.
1 Introduction and main results
We consider the system
! !" $#!"
% & (') * +, !" -#!"
.
(1)
where /1032465
87 , 9 : 9 !"<;;;<"
9 2
#>=?
2 , @ =3ACBD"-EFA
, and moreover G !" $#%
!" $#H
9 " @
#%=)?
. Further, IJB and K=)? are parameters, whereas 4 $" !# denote the nonlinearities. The functions 4 L 4 9 # are supposed to be spatially localized, i.e., they decay to zero at some exponential rate as M9NMPO E . Throughout this paper we assume the functions 4RQ ? 2 O ? and the nonlinear terms +S Q ?
O ?
UTV:W"-X#
to be at least Y times continuously differentiable, withY =[Z satisfyingY I]\_^X .
Our assumptions will be such that for =a`
b?
2
"c?
#
the stationary solution dB of (1) is only marginally stable, in the sense that the associated linearizationefcgh about FB
possesses essential spectrum up to the imaginary axis. This operatore fcgh h 9 # splits into two parts, namely into an 9 -independent part and into an 9 -dependent part
h 9 #
, respectively given by
3
# and h 9 #
,
9
#H
9 #
'<
9 # ,
9
#
; (2)
According to a classical result (cf. [10, Theorem A.1]) the essential spectrum of the operator
is ' M M
KT Q
$";<;;<"
2 # = ?
2 , and it is not affected by adding the relatively compact perturbation h 9 # . However, a number of isolated eigenvalues could be created through the operator h 9 # . In order to formulate our precise assumptions, we introduce a spatial weight which results in a shift of the essential spectrum into the left half plane, whereas the isolated eigenvalues remain unchanged. For = ? we define the operator egh by
egh
e fcgh
#$"
i.e., we have
egh
L
'X
#
c
'
#
h 9
#c#
"
(3) with denoting identity. The spectrum of egh consists of essential spectrum to the left of
= Q
' and of a number of isolated eigenvalues.
Now we are ready to state our main hypotheses. Here and henceforth we fix )I B and choose H^X . Since we consider as the bifurcation parameter we will mostly suppress
in our notation and just write e h instead ofe,gh . (H1) There exists ! IKB such that
M 4 H
9 # M M 4
<,
9 # M"#!
%$'&
&( " M)*M"KY
" 9 = ? 2 ;
(H2) There exist ! IB and* = Z with*,+ X such that for all =?
with M M-" W we have
M
) #
M".! M M/ "
where ) #( U< $" !#$"-+ !" $#c#$;
In addition,) # # for =
.
(H3) The functionsH 9 # and 9 # are chosen in such a way that the following holds. There exist
$
=a? , 0Hf I/B , and 1 I B such that for =32
$ ' 0Hf
"
$ 0Hf
A all eigenvalues 4 =5 of e h , except of two, satisfy 46" ' 1 . The other two eigenvalues
487
f
# satisfy
4 7f $ # 9%T;:
f=<
B
(4) with: f IKB , and
\
\_
487f U&#>
>>h 5
h@?
IKBD;
(5)
Examples of functions and such that (H1) and (H3) hold are 4 9 #Va 4BACEDGF
U\ 4 M9NM #
for) JW"$X , with constants 4 and\ 4 properly chosen.
Remark 1.1 From assumption (H3) we have the following consequences for the spectrum of
e,gh for a variable = ABD"-H^X2
. As mentioned before, the spectrum ofe gh consists of essen- tial spectrum to the left of = Q
' and of a number of isolated eigenvalues.
If f is an eigenfunction ofe
$;(
gh with eigenvalue4Pf , then a straightforward calculation us- ing (3) shows that H 9 #+
%$;(
f 9 #
is an eigenfunction of e,gh corresponding to the same eigenvalue4 f . Therefore the isolated eigenvalues ofe gh are independent of until they vanish in the essential spectrum, cf. [13].
For future applications we also consider another class of amplifications and nonlinear terms which are of Navier-Stokes type.
(H1)’ Let 9 # 9 # for W"$X and some = W";;;!"-\
. There exists! I B such that
M 4 H
9 # M M 4
<,
9 # M"#!
%$'&
&( " M)*M"KY
" 9 = ? 2 ;
(H2)’ There exist! I]B and* = Z with* + X such that
) # 2
4c5
87
4 #
and for all = ?
with M M%" W that
M 4 #
M%".! M M/ "
where
4
#( U
4
!" $#!"
4
$" !#c#$;
In addition,
4 # 4 #
for =
.
Using the incompressibility condition, the nonlinear terms of the Navier-Stokes equations can be cast into the form considered in (H2)’. In contrast to (H2)’ it is not obvious whether the technical assumption (H1)’, which is needed in Section 3 and which implies (H1), can be satisfied in applications. See also Section 4.
Assumption (H3) is reminiscent of the assumption for the classical Hopf bifurcation, and so the purpose of this paper is to investigate the bifurcation scenario of (1) in a neighborhood of B for close to
$
. The new difficulty which occurs here is due to the fact that the linearizatione fcgh about FB possesses continuous spectrum up to the imaginary axis, without any spectral gap, as is indicated in Figure 1.
Therefore the nonlinear stability of LB for]
$
, the occurrence of a Hopf bifurcation at
J
$
, and the exchange of stability from 3B to the bifurcating time-periodic solutions are not clear at all.
Our first theorem concerns the stability of the trivial solution B for3
$
. We prove the nonlinear stability with respect to spatially localized perturbations.
Theorem 1.2 Assume I B is fixed, and (H1), (H2) or (H2)’, and (H3) are satisfied. If
*
IJW ]X^\
for (H2) or if* I X^ \ for (H2)’ and
$
, then there exists I B such that for every0 I]B one may choose 0 VI B with the property that
"-B #
"-B #
"-B #
".0
Re essential Im
eigenvalues discrete
Figure 1: Spectrum of the linearizatione fcgh about the trivial solution LB
for the initial data implies
A
fcg W(
@ # 2 ( " @ #
" @ #
,
" @ #
".0
for the solution of (1), where 9 " @ # 9 " @ #
with <^ X .
The proof, which is elaborated in Section 2, is based upon the following facts. The term
h 9 #
in (1) leads to an amplification of perturbations of B near 9 B . This is counter-acted by the effect of the drift term
_
which tends to transport these perturba- tions away from 9 dB to infinity. Thus after a sufficiently long time the diffusion term will dominate and force the perturbations to decay with rate @ 2 (
. Generally speaking, all nonlinear terms are asymptotically irrelevant with respect to diffusion, i.e., the nonlinear system
) #!"
M5 f "-B #$"
with: U!"-+-#
shows the same asymptotic behavior as the linear diffusion equation, for sufficiently small and sufficiently spatially localized initial data "-B# ; see e.g. [3]. From a technical point of view, to recover the diffusive behavior behind the amplification we intro- duce suitable norms with exponential weights in space. For perturbations which are small in these norms the influence of the spatially localized amplification vanishes exponentially in time, since the distance between the main perturbation and the spatial amplification at9 LB grows linearly; see [6] and the references therein.
At
$
two complex conjugate eigenvalues with nonvanishing imaginary part cross the imaginary axis. Hence the trivial solution becomes unstable. Although we have continuous spectrum up to the imaginary axis a finite dimensional reduction is possible, and like in the case of a classical generic Hopf bifurcation the problem of bifurcating time-periodic solutions is reduced to the solution of an equation of the form
' $
" #N < U[' $ #
F ; ;; LB "
with a smooth function
Q ? O ?
and coefficients 4 = ? related to the nonlinearity;
cf. Remark 3.15 below for more precise information, in particular for the definition of the coefficient . Depending on the sign of the next theorem guarantees the occurrence of a supercritical or a subcritical Hopf bifurcation. It is formulated only for the non-degenerated case, i.e., LB< .
Theorem 1.3 Assume>ILB is fixed, < B , and (H1), (H2) or (H1)’, (H2)’, and (H3) are satisfied. If\ + for (H1), (H2) or if\ + W for (H1)’, (H2)’, then (1) has a one-dimensional family of small time-periodic solutions, i.e., there exists f IKB such that for all =32 B " f A we have
9 " @ #
9 " @ X^ : #
solving (1) for
$ ' A !#
. Moreover,:] : f # , and
fcg
(!
#"
#
. If in addition the hypothesis
(H4) System (1) is equivariant (cf. [8]) under
%$
O '
holds, then the assertion of the theorem is true regardless of the space dimension\ + W .
The proof is given in Section 3. The periodic solutions can be written as convergent series
&
9 " @
#(
'
)(
' 9
# S' "
where for * dB< the ' vanish with some uniform (in * ) exponential rate as M9NM O E , whereas in general f =,+.- is not decaying. However, in the important special case that the equivariance condition (H4) is satisfied, we have f B on symmetry reasons; see Lemma 3.1 below. An example of a system (1) which is equivariant in the sense of (H4) is provided by the nonlinearity
H !" $# ' M M M M #
and !" $# ' M M
M M
#!"
(6) which also satisfies (H2) with* 0/ , whence Theorem 1.3 applies.
The proof of Theorem 1.3 is again based on an interplay of the (spatial) uniform norm and an exponentially weighted norm. This interplay allows us to use the classical Lyapunov- Schmidt method. The difficulties with the continuous spectrum touching the imaginary axis then become evident in an equation for f (the mean value in time) to be solved, which has the form
f
c
f
21
in the case of (H2), and which reads as
f
c
f 87
1
in the case of (H2)’. Although the spectrum of the operator on the left-hand side touches the imaginary axis, these equations admit a unique solution, if (H2) holds in at least four space dimensions, respectively if (H2)’ holds in all space dimensions. In the Lyapunov-Schmidt
procedure the function 1 will contain the nonlinear terms, and due to the Cauchy-Schwarz inequality the additionally needed assumption1]=K`
is satisfied. According to the above remarks the difficulties in solving these equations do not occur if (H4) is assumed, since then
f B
.
The presence of a time-independent part f of makes the proof of the exchange of stability from the trivial solution /B to the time-periodic solution
more delicate.
As mentioned before, the time-dependent parts of decay with some exponential rate in space. This allows us to handle these terms once more by introducing a suitable exponential weight in space, and hence in the case that f LB the exchange of stability works as for the classical Hopf bifurcation. However, the time-independent part f in general only decays at a polynomial rate and cannot be dealt with in the same manner. But nevertheless the part associated to f is still a relatively compact perturbation which can be handled by well known estimates for linear Schr¨odinger operators. These estimates are very similar to the estimates which we use in the proof of Theorem 1.2.
Hence the classical exchange of stability can be established if ]B .
Theorem 1.4 Assume IJB is fixed, FB , and (H1), (H2) for\ + or (H1)’, (H2)’ for
\ + X
, and (H3) are satisfied. Then there existsf I B such that for all =32
$
"-
$ f
A the time-periodic solution 9 " @ # is stable with respect to spatially localized perturbations.
More precisely, forK=32
$
"-
$ ,f
A there exists f I3B such that for all =-2BD" f A there is
I B
such that the following is true. For all 0 I B there exists 0 I B such that for the solutions of (1) we have
A
fcg
8W @ # 2 ( " @ #
R " @ #
" @ #
".0
!"
provided the initial data satisfy
"B #
R "-B #
"-B#
" 0
"
where 9 " @ #NR 9 " @ #
.
The proof is carried out in Section 4.
There are a number of physical problems where the situation occurs which is considered here. Examples are the flow around a body in the Navier-Stokes equation [1, 2] or the bifurcations of a tip of a spiral wave [18]. It is the purpose of this paper to come closer to a mathematical understanding of the bifurcation scenario in such systems. The bifurcation scenario is delicate due to the fact that for all values of the bifurcation parameter we have essential spectrum up to the imaginary axis. Therefore classical reduction methods (which are available if the spectrum of the linearization separates into a finite-dimensional part close to the imaginary axis and an infinite-dimensional part which lies strictly in the left half-plane of the complex plane) as the Lyapunov-Schmidt method and the center manifold theorem are no longer available, at least in the classical set-up.
i) stable
unstable stable
α α
unstable
unstable stable
ii)
Figure 2: Exchange of stability in the classical and diffusive Hopf bifurcation, i) if B and ii) if IKB .
As a first step towards an understanding of this question we considered in [15] a nonlin- ear diffusion equation in one space dimension, in the case of a real eigenvalue crossing the imaginary axis. There we proved the nonlinear stability of the trivial solution with respect to spatially localized perturbations if there is no eigenvalue in the right-hand half plane, the oc- currence of a pitchfork bifurcation in case of the real eigenvalue crossing the imaginary axis, and finally an exchange of stability from the trivial solution to the bifurcating equilibrium.
Investigations of bifurcation scenarios including continuous spectrum can also be found in a number of other papers, see for instance [4, 9, 11, 12, 14, 16, 19] and the references therein.
Notation. Throughout the paper different irrelevant constants are denoted by the same sym- bol ! . For simplicity we often write
, although = ?
. Here
stands for any of the terms
with . Mostly we will be not too precise about the particular form of the nonlinearity , since this would lead to much additional notation. However, for definiteness always the nonlinearity from (6) should be kept in mind.
Acknowledgement. Guido Schneider would like to thank Jean-Pierre Eckmann and Peter Wittwer who helped by useful comments to prove the ”estimates on ” which are basic for the exchange of stability in Section 4 with non-vanishing “background” f . This argument was completed while Guido Schneider visited the Theoretical Physics Department at the University of Geneva whose hospitality is gratefully acknowledged. Moreover, he would like to thank Thierry Gallay and Christian Simader for interesting discussions.
2 Nonlinear stability of the trivial solution for
It is the purpose of this section to verify for K
$
the stability of the trivial solution LB with respect to spatially localized perturbations. Although only a suitable modification of the proof given in [15, Thm. 2.1], the details are included since the method will be used again in Section 4. However, for simplicity we will restrict ourselves to nonlinearities satisfying (H2).
The result will be established by simultaneously controlling the solution of (1) in three different norms. In order to estimate the influence of the amplification, we rely on a weighted norm in space and introduce 9 " @ #N 9 " @ #
, with <^ X . Then (1) and (3) yield
e h
"
#!"
with " #N#
) #
being a smooth nonlinear mapping, where we understand that in
) #
each factor could be replaced by
if desired. For instance, if is a polyno- mial, then we replace every
in) # by
, which results in the contribution
toR " # . In particular, (H2) yields
M 9
#$"
9
#6#
M"#! M 9 # M/ M 9 # M;
(7) The choice of is not unique, but for the rest of the paper just one such is fixed.
Example 2.1 For the nonlinearity from (6) we let
"
#&
'
<
M M M M #
' ,
M M M M #
for a !" -# and !" $# .
Conversely, the variable will be used to replace in (1) where it appears with an exponen- tially vanishing factor; at this point we use the fact that h decays to zero at an exponential rate as M9NM O E . Hence we are led to consider the augmented system
h 9 #
) #!"
e h
"
#!;
.
(8)
For this system we need to derive bounds both in the
-norm and in the
-norm;
we make this evident in our notation by denoting the same solution by , if =
! f
! f
? 2
"6?
#
, and by , if =K`
1`
b?
2
"c?
#
. Then , , and are a solution of the system
h 9 #
#!"
h 9 #
H
"
#!"
e h
"
#!"
(9)
where < " # is a smooth nonlinear mapping which is obtained from ) # in the same way as " # is obtained from ) # , e.g.,
in ) # will be replaced by
. Hence we also have
M
!
9
#$"
9
#c#
M".! M 9 # M/ M 9 # M;
(10)
essential Im
eigenvalues discrete Im
Re Re
Figure 3: Spectra of the linearizations
h 9
#' and egh in (8) about the trivial solution B .
Although not explicitly stated we will also use the -variable in the first equation of (9). To get a clue on what kind of behavior may be expected for the diffusive variables and , we note that " @ #N '[ @ !" @ # and " @ #N '[ @ !" @ # satisfy (with W"-BD";<;;H"B #c#
the system
#:;;;"
H
"
# ;;<;"
.
(11) the dots indicating terms that vanish at an exponential rate. For (11) it is known that the diffusive behavior of the linearized system can be used to control the nonlinear terms and to prove diffusive behavior also for the nonlinear system. Hence we expect, for sufficiently small spatially localized perturbations, the -variable in (9) to decay as @ 2 (
#
, the - variable to remain bounded, and accordingly the -variable to decay at an exponential rate.
These arguments are made rigorous in the following Proof of Theorem 1.2 : We introduce
@
# A
fcg
8W # 2 ( # "
@
# A
fcg
#
"
@
# A
fcg
#
"
where # denotes " # , etc., and ILB is specified below in (13). From the variation of constants formula and (9) we obtain
@ #
f
f
h 9 #
#)
#c# \ "
@ #
f
f
h 9 #
#<
#!"
#6#
\ "
@ #
f
f
A R
#!"
#c# 2 \ "
with f 9 #
f 9 #
. Next we remark that
"#!
"
"#!%@
2 ( "
"#!
"
(12)
".!
"
for suitable constants ! I B and I B . The first three estimates are due to the fact that the solution of ] with initial dataNM5 f f is given by 9 " @ #V
9
@
!"
@ #
, where a W"-BD"<;;;!"-B #
and
9 " @ # W
X!
@ # 2 ( !
"$#
f &% #8\'%P;
Further, the operatore h is sectorial. Hence the bound on
is obtained with
W
/)(+*
-,
^ " 1 " M
4/.
f
&#
M" M 4
f
U&#
M10 IKB "
since the essential spectrum of e h lies to the left of = Q3 '] 'V ^ , and by (H3) the eigenvalues except of two are located to the left of = Q ' 1 , and for the remaining two eigenvalues we moreover have 4 7f &# ]B by (4) and (5). With this choice of we moreover set
;
(13) Moreover, we note that the third estimate of (12) is not needed for all @ + B , but only for
@ = ACBD"<WE2
.
First we estimate
@ #
" f
f
R
#!"
#c#
\ ;
As a consequence of (7) we see that M 9 " #$" 9 " #c# M".! M 9 " # M/
M 9 " # M, and this yields
R
#!"
#c#
".!
# /
#
. Hence we obtain
@ #
"6!
f
!
f # /
#
\
"6!
UB #
! @ #/ @ #
f W # 2 / ( \
"6!
UB #
@ #/ @ # ;
Taking the supremum over time, it follows that
@ #
".!
B#*
@ #/ @ # "
(14) and here no assumption on* + W is needed, since all nonlinear terms can be controlled by the exponential decay of the linear semigroup
. Next we invoke (H1) and <^X to get M
h 9 #
M%".!
$'&
&(
%$'&
&(
".! , and thus (10) yields
@ # "
Pf
f
h 9 # # \
f
H
#$"
#c#
\
" ! UB #
!
f #
\ !
f # /
#
\
" ! UB #
! @ #
f \ ! @ #/ @ #
f
8W # 2 / ( \ ;
Therefore
@ #
".!
B #
@ # / @
#
@ # ;
(15)
