Hua C., Wong R. Differential Equations and Asymptotic Theory in Mathematical Physics
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92
precisely to their
local
study). These tools work for solutions but
also
for other classical and important problems
as
normal
forms.
These are
important results in this direction for the case of (germs of) analytic differ-
morphisms tangent to identity:
p(z)
=
z
+.
.
.
(G.D. Birkhoff, Malgrange,
Ecalle, Voronin, more recently Mardesic and Rousseau
. . .
)
and for the
case of
resonant
(germs of) analytic 0.D.E in the plane (Martinet, Ramis,
and more recently
L.
Teyssier, Voronin,
.
. .
)
or in bigger dimensions
(L.
Stolovitch, B. Braaksma
. .
.).
In such situations we have
convergent
nor-
mal forms and Gevrey, or more precisely k-summable coordinate transfor-
mations reducing a systems to “its” normal form. Comparison between the
natural sums gives rise to
non-linear Stokes phenomena
[12].
Resonant differential equations in the plane are local perturbation
of
xCk+ldy
+
Xydx
=
0
(saddle-nodes) or
pxdy
+
qydx
=
0
(p,
q
E
N+).
When
we perturb
a
nilpotent equation
as
d(y2
-
x3)
=
2ydy
-
2x2dx
=
0,
the
situation is more dedicate. We have “prenormal forms” in Takens style.
Some are convergent (a recent result of Zoladek). But the “true” normal
forms (due to
F.
Loray) are
divergent.
M. Canalis-Durand and R. Schafke
recently proved that they are in fact Gevrey and even summable
[7].
The
same authors and
E.
Paul are working on generalizations.
Another potential field of applications is the field of
singular pertur-
bations
of analytic dynamical systems. Gevrey asymptotics theory is the
asymptotic theory
exactly adapted
to this field which is extremely impor-
tant for the applications.
This appears clearly in the recent
complete solution
by A. Fruchard and
R. Schafke of the famous problem of Ackerberg-O’Malley resonance
[9],
[ll]
or in the recent solution of the important Wasow’s conjecture for
turning
points
by Catherine Stenger
[32].
In such problems it is clear that the
good theory
is
Gevrey asymptotics
(not PoincarB’s Asymptotics which is
too unprecise,
or multisummability which is
“too precise”
. . .
,
except for
some particular situations
),
4.1.
Delay
in bifurcations
We begin with discrete dynamical systems. We study the iteration of a
map
f
:
R
+
R
or
C
4
C,
i.e the family of maps
(composition
n
times;
n
E
N).
For sake of simplicity we will suppose that
F
is polynomial;
F
E
R[x]
or
@[XI.
More generally we can consider a
family
of
93
maps
{FA}AE~.
We can “slightly” modify the problem, looking at a “small
perturbation” of this situation.
Let
E
>
0
a
“small” real numberlg. We
replace the family of iterations:
(z,
A)
H
(FA(%)
=
f(z,
A), A)
by
(z,
A)
I-+
FE(X,
A)
=
(f
(z,
A),
A
+
E)
(we change
a
little bit the real (or complex) parameter
A
at
each iteration
map). We will investigate what happens when we perturb a
bifurcation
of
the family by
a
slow
drift.
A
typical example is the Feigenbaum family:
f(z,
A)
=
Az(1
-
z).
We need a hypothesis: we will suppose that there exists an analytic (in
A)
curve
of
fixed points
A
I-+
Co(A)
for the unperturbed family.
This problem has been studied by
C.
Baesens (by a Gevrey approach)
and, independently, A. F’ruchard (using Neishstadt ideas in the spirit of the
summation at the smallest term).
An invariant curve for the perturbed family is
A
H
U(A,
E)
with
F€(U(A,E),A)
=
U(A+E,E).
The
two
authors prove the existence of such a curve if
d,F,(A,
co(A))
#
1
formally:
these exists a
unique
B(A,
E)
=
%(A)
+
c
Cn(A)P
n>
1
satisfying our relation. In general
0
is divergent but2’ Gevrey
1
in
E
(the
c,’s
being holomorphic and uniformly bounded in
n
on a same disc in
A).
By an incomplete Laplace transform we can get
a
“quasi-invariant quasi
curve” (an exponentially narrow “invariant domain” at a distance of order
In Fig
4.1.1,
the dotted lines correspond to the bifurcation of equilib-
riums for the unperturbed family. The black line corresponds to the same
bifurcation for the drift case. We observe a
delay
in
the bifurcation,
Feigen-
baum drifted case is very interesting. The reader can easily draw the cor-
responding drifted LLcascade” of bifurcations. This drifted case appears in
some laser problems.in physics. The explanation of the delay phenomenon
E
of
Q).
IgIn some works on this subject,
E
is
“infinitely small” (Non Standard Analysis).
‘OC.
Baesens.
94
I
//w//
I
>parameter
1
Fig
4.1.1
is essentially the same in the discrete and continuous case. We will limit
ourselves to the continuous case.
We consider a family of vector fields on a domain of
@P
(or
RP).
The
parameter
X
is complex (or real):
.
dx
dt
2
=
-
=
fx(x)
(x
E
CP)
is the corresponding family of autonomous
0.D.E
(t
real or complex is the
“time”). We will always suppose that
fx
is analytic in
x
and
A.
We now consider a small perturbation (.a
non autonomous
one),
E
>
0
being
a
fixed
small real;
af
We can suppose that
f
(0,O)
=
0.
The simplest case is when
-(O,
0)
E
GL(n;@)
(2.e.
when the linear part is
invertible
near the origin). Elimi-
nating the time
t,
we get
ax
dx
dX
&-
=
f
(X,x).
d
dX
The operator
E-
is a
singular perturbation;
if
E
=
0,
we get
0
which is not
differential. The discrete case
X
H
X
+
E
corresponds to exp(&-) which
can be considered as a singularly perturbed operator, differential
of
infinite
order.
d
dX
95
dx
dX
For
E
=
0,
E-
=
f(X,x)
become
0
=
f(X,z).
This is an analytic
implicit
equation. With our invertibility hypothesis, it admits (in a neigh-
borhood of
X
=
0)
a unique solution
co(X)
which is analytic:
n>O
We will suppose that the so-called
slow curve
X
H
co(X)
is tangent to the
field
(i.e.
co
#
0)
dd
ax
dX
f
(5,
A)-
+
E-.
During the spring 1989, in relation with a phenomena of delay of bifur-
cation observed experimentally for such families, the French mathematician
J.
Martinet conjectured the following.
Conjecture
4.1.
(Martinet)
With
the above notation and hypothesis, there
exists
a
unique
formal
curve
E(E,
A)
=
C
invariant by the field,
where the
en’s
are holomorphic and uniformly bounded
in
a same disc.
Moreover,
2
is Gevrey
1
in
E
(uniformly
in
a suficiently small
E).
n20
We suppose that when
X
varies there is a stable equilibrium
for
X
>
0
(A
E
R)
and that we loss the stability for
X
=
0:
we have an unstable
equilibrium for
X
>
0
(typically a Hopf bifurcation at
A
=
0
with the birth
of
a
limit cycle growing when
X
growth,
X
>
0).
In such a case we can
observe a delay in the bifurcation for the perturbed case: the bifurcation is
observed
for
a
strictly positive
value of the parameter
A.
J.
Martinet gave an explanation of this phenomena based upon his con-
jecture.
Gevrey estimates
are
essential
in this approach.
From Conjecture 4.1 we get a
“quasi-invariant quasi-curve”:
we use an
incomplete Laplace transform
of
2
(in the variable
E).
We get
a
“quasi-inva-
riant curve”
(quasi-curve because of the cutting choice
. . .
).
It
is important
to notice that in general
E
is divergent in
E.
Roughly speaking we have an
invariant region which is exponentially narrow (in
E).
The distance between
co
and this quasi-curve is of the order of
E
(-
c~E),
the concentration is of
the order of some
e-‘lE
(typically
E
=
-
C~E
N
-,
e-‘/‘
=
e-k).
We can apply a linearization of our 0.D.E “along the quasi-invariant
curve” (Poincar6 variational equation). Up to
exponentially small errors
1 1
10
l
5
96
we are reduced21 to the case of
a
curve which is analytic and invariant by
the field. This case is similar to the following case: while a slow curve is
invariant by the field
CO,
it is easily reduced to
co(X)
E
0.
Then the explanation of the delay
of
bifurcation is evident. We will
see that on a simple example. We consider the simplest example of our
situation:
x
=
px
{
fi=E
(4.1.1)
Here
X
=
p.
For
p
#
0
fixed,
we consider the
0.D.E:
x
=
px
(4.1.2)
The point
x
=
0
is an equilibrium for (4.1.2).
It
is a stable equilibrium for
p
<
0
and an unstable one for
p
>
0.
There is a
bifurcation
for
p
=
0.
We
consider the unique solution of (4.1.1) corresponding to the initial condition
x
=
20,
p
=
po
<
0.
We see immediately that it can be computed “in closed
(4.1.3)
(ro+.tP
-2
(From 4.1.3 we get
x(~,
t)
=
xoe
2~
.)
We see that for
p
E]
-
po,po[,
X(E)
is infinitely small in
E.
Then our
solution goes down nearly vertically from the initial condition
(XO,
po)
to a
point exponentially near
(0,
PO),
follows the “horizontal” real axis until
p
arrives exponentially near
of
-po
>
0,
then jumps almost vertically to the
“final condition”
(20,
-PO).
When the solution follows the horizontal axis
it remains exponentially near
of
this axis (cf. Fig 4.1.2).
It
remains to prove Martinet conjecture (conjecture 4.1). It was proved
by
Y.
Sibuya almost at the same time it was stated by
J.
Martinet (and
completely independently from any consideration of delay in bifurcation!)
Theorem
4.1.
[Sibuya) Let
n
E
N*.
We consider a differential equation
where
CT
E
N*,
E,
x
E
C,
y
E
C”.
We suppose that
F
is analytic
in
(x,
E,
y)
in
a neighborhood
of
(0,
0,O)
and that
f
(0,O)
=
0.
We suppose also that the
matrix
A(0,O)
is invertible.
’lMore precisely
the
two
cases
are
similar.
97
I
Fig
4.1.2
Then the equation admits a unique formal (in
E)
solution y
=
f
(2,
E)
=
an(x)En, where the
an’s
are holomorphic and bounded
in
a same disc
n>O
1
of the x-plane. Moreover (reducing this disc
if
necessary),
f”
is
Gevrey
-
in
E:
U
Sibuya’s proof
of
his theorem was delicate.
It
used cohomological esti-
mates and Ramis-Sibuya theorem. Later M. Canalis-Durand gave a direct
proof using a Gevrey method for
P.D.E
(“MQthode des contours succes-
sions”). Later
R.
Schafke gave a very simple
proof
using the same trick
(Nagumo norms) and a majorant equation (which is a singular
0.D.E).
d
d
dx
dz
(The idea is to “replace”
E-
by
z2-.)
Now there is
a
natural question: what happens if in Sibuya theorem,
or Matinet conjecture
.
.
.
.
we suppress the invertibility condition?
An
example of such a situation was studied
before
Martinet work, it is the
so-called
duck phenomena
(Ye phQnomQne canard”).
We will describe this phenomena, more precisely how it was discovered
by
J.L.
Callot,
F.
and M. Diener and
E.
Benoit following an idea
of
the
French (Alsacian
. .
.)
mathematician G. Reeb
at
the beginning of the
eighties
[4].
Later we will see that it is related to
a
more sophisticated
version
of
Sibuya theorem
[6].
But that was discovered
a
long time after!
Finally we have an unified theory involving
a
lot of Gevrey asymp-
98
totics”.
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1.
W.
Balser,
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SPIKES FOR SINGULARLY PERTURBED
PROBLEM
REACTION-DIFFUSION SYSTEMS AND CARRIER’S
MICHAEL
J.
WARD
Dept.
of
Mathematics,
University
of
British Columbia,
Vancouver,
B.C.,
Canada V6T
122
E-mail: wardamath. ubc. ca
1.
Introduction
For a two-component reaction-diffusion system, Turing
[97]
in
1952
used
a
linear stability analysis to show that
a
spatially homogeneous steady-state
solution that is stable in the absence of diffusion can be destablilized when
the ratio of the diffusion coefficients is sufficiently large. This diffusion in-
duced instability of the homogeneous state leads to various types of stable
spatial patterns. Spike patterns are those where one or both of the chemical
species concentrate,
or
localize, at certain points in the domain. There has
been considerable progress over the past thirty years in developing weakly
nonlinear theories to analyze small amplitude patterns that emerge from
an initially spatially homogeneous solution. However, there are many open
problems that relate to the stability and dynamics of fully nonlinear, but
localized structures, in reaction-diffusion systems. In the survey chapter
on pattern formation by Knobloch (see
[61]),
he remarks that “The ques-
tion of stability
of
finite amplitude structures, be they periodic or localized,
and their bifurcation is a major topic that requires new insights”. In this
survey we make an effort in this direction by giving some results for the ex-
istence, stability, and dynamics,
of
spike patterns to a few reaction-diffusion
systems, including the Gierer-Meinhardt (GM) model
[36],
the Gray-Scott
(GS) model
[37],
and the Schnakenburg
(SC)
model
[go].
Specific open
problems are also mentioned.
100
101
The GM model has been used to model a range of localization processes
in nature, including morphogenesis and the formation of sea-shell patterns
(cf. [70], [71]). In dimensionless form, the GM model is given by
ap
at=E2Aa-a+-,
XER;
dna=O,
XE~R,
(l.la)
Tht
=
DAh
-
k+
-
XER;
d,h=O,
XE~R.
(1.lb)
hq
am
cNhs
’
Here
R
is
a
bounded simply-connected domain in
RN,
a
and
h
are the acti-
vator and inhibitor concentrations,
c2
<<
1
and
D
are the two diffusivities,
T
is a reaction-time constant, and
3,
denotes the outward normal derivative.
The exponents
(p,
q,
m,
s)
in the
GM
model are assumed to satisfy
p>l,
q>O,
m>0,
s>O,
<=--
mq
(s+l)
>
0.
(1.2)
For
N
>
3, we assume that
p
<
p,
=
(N
+
2)/(N
-
2),
where
p,
is the
critical Sobolev exponent.
The Gray-Scott system, introduced in [37] and popularized in the nu-
merical study of [86], models an irreversible reaction involving two reactants
in a gel reactor, where the reactor is maintained in contact with
a
reservoir
of one of the two chemicals in the reaction. In the nondimensional variables
of [74], this system can be written in one spatial dimension
as
P-1
(1.3a)
W~=E
2
w,,-v+Auw~,
JzJ<l;
~,(fl,t)=O,
=
Du,,
+
(1
-
U)
-
UW~,
1x1
<
1
;
~,(fl,
t)
=
0.
(1.3b)
Here
D
>
0,
T
>
1,
E
<<
1,
and
A
>
0.
Two ranges of the feed-rate
parameter
A
are of interest: the high feed-rate regime where
A
=
O(1)
and the low feed-rate regime where
A
=
O($12).
In this latter regime, we
introduce the new variables
In terms of (1.4), (1.5) is transformed to
(1.5a)
(1.5b)
2
vt
=
E
v,,
-
u
+
2uv2
,
121
<
1
;
u,
(51,
t)
=
0,
TUt
=
Du,,
+
(1
-
U)
-
E-~uu~,
1x1
<
1
;
~,(fl,
t)
=
0.
The Schnakenberg model
I901
is another well-known two-component
reaction-diffusion system. In non-dimensional form, and in one spatial di-