Bifurcation delay - the case of the sequence: stable focus - unstable focus - unstable node

1.7 MB·English

by  Eric Benoît

Bifur
ation delay - the
ase of the sequen
e:
stable fo
us - unstable fo
us - unstable node
∗
Eri
Benoît
January 20, 2009
9
0
0
2
Abstra
t

n
Letusgiveatwodimensionalfamilyofrealve
tor(cid:28)elds. Wesupposethatthereexistsastationarypoint
a
J where the linearized ve
tor (cid:28)eld has su

essively a stable fo
us, an unstable fo
us and an unstable node.
When the parameter moves slowly, a bifur
ation delay appears due to the Hopf bifur
ation. The studied
9
1 question in this arti
le is the
ontinuation of the delayafter thefo
us-node bifur
ation.
AMS
lassi(cid:28)
ation: 34D15, 34E15, 34E18, 34E20, 34M60

] Keywords: Hopf-bifur
ation, bifur
ation-delay,slow-fast,
anard, Airy,relief.
S
D
. 1 Introdu
tion
h
t
a "Singularperturbations"isastudieddomainfrommanyyearsago. Sin
e1980,many
ontributionswerewritten
m
be
ause new tools were applied to the subje
t. The main studied obje
ts are the slow fast ve
tor (cid:28)elds also
[ known as systems with two time-s
ales. We will give the problem here withεaX˙m=orfe(pt,aXrt,i
εu)lar poinεt of view:
the bifur
ation delay , as in arti
les [8, 2, 9, 7℄. We write the studied system: , where is a real
1
v positive parameter whi
h tends to zero. For a better understanding of the expression dynami
bifur
ation it is
3 better to write the system after a res
aling of the variable:
8 X˙ = f(a,X,ε)
8
a˙ = ε
2
(cid:26)
.
1 a
0 where is a "slowly varying" parameter. X˙ = f(a,X,0)
9 The main obje
ts in this study are the eigenvalues of the linear part of equation near the
0
quasi-stationarypoint. Indeed, they givea
hara
terizationof the stability ofthe equilibrium of the fast ve
tor
:
v (cid:28)eldatthispoint. Theaimofthisstudyistounderstandwhathappenswhenthestabilityofaquasi-stationary
Xi point
hanges. A bifur
ation o

urs when at least one of the eigenvalues has a null real part.
Inthisarti
lewerestri
tourstudytotwo-dimensionalrealsystems. Inthissituation,thegeneri
bifur
ations
r
a are: the saddle-node bifur
ation, the Hopf bifur
ation and the fo
us-node bifur
ation.
Thesaddle-nodebifur
ationissolvedbytheturningpointtheory: whentherealpartofoneoftheeigenvalue
be
omes positive, there is no delay and a traje
tory of the systems leaves the neighborhood of the quasi-
stationary point when it rea
hes the bifur
ation. For this study, the study of one-dimensional systems is
su(cid:30)
ient: we have a de
omposition of the phase spa
e where only the one-dimensional fa
tor is interesting.
There exist many arti
les on this subje
t, we will be interested parti
ularly by [3℄ where the method of relief is
used. The arti
le [6℄ introdu
es the geometri
al methods of Feni
hel's manifold.
The Hopf delayed bifur
ation is well explained in [10℄, we will upgrade the results in paragraph2 below.
In a fo
us-node bifur
ation, the stability of the quasi stationary point does not
hange, then, lo
ally, there
is no problem of
anards or bifur
ation delays. Indeed, when there is a bifur
ation delay at a Hopf-bifur
ation
point, it is possible to evaluate the value of the delay, and the main question is to understand the in(cid:29)uen
e of
the fo
us node bifur
ation to this delay.
In paragraph2, the Hopf bifur
ationaloneis studied, as well asthe fo
us-node bifur
ationfollowingaHopf
bifur
ation in paragraph3.
∗
Laboratoire de Mathématiques et Appli
ations, Université de la Ro
helle, avenue Mi
hel Crépeau, 17042 LA ROCHELLE
and/or projet COMORE,INRIA,2004route desLu
ioles06902SOPHIA-ANTIPOLIS
ourriel: ebenoituniv-lr.fr
1
In paragraphs 2.1 and 3.1, we assume that there exists a solution of the system approximed by the quasi
steady state in the whole domain, so this traje
tory has an in(cid:28)nite delay. The used methods are real, and the
C2
system has to be smooth (a
tually only ). In paragraphs2.2et 3.2, we avoidthis very spe
ial hypothesis. It
is here supposed that the system is analyti
, and we study the solutions on
omplex domains. Unfortunately, I
havenot a proof for the main result of this arti
le. But it seems to me that the problem is interesting, and the
results are argumented.
WeuseNelson'snonstandardterminology(seeforexample[5℄). Indeed,almostallsenten
es
anbetranslated
ε
in
lassi
al terms, where is
onsidered as a variable and not as a parameter. Often, the translation is given
on footnotes.
2 The delayed Hopf bifur
ation
The problem is studied and essentially resolved in [10℄. We give here the proofs to improve the results and to
(cid:28)x the ideasfor the mainparagraphof the arti
le. The main toolis the relief's theory ofJ.L. Callot,explained
in [4℄.
The studied equation is εX˙ =f(t,X,ε)
(1)
f 2
where is analyti
on a domain D of C×C ×C.
Hypothesis and notations
f
H1 The fun
tion is analyti
. It takes real values when the arguments are real.
ε
1
H2 The parameter is real, positive, in(cid:28)nitesimal .
φ f(t,φ(t),0)=0
t
H3 There exists an analyti
fun
tion , de(cid:28)ned on a
omplex domain D so that . The
urve
X = φ(t)
t
is
alled the slow
urve of equation (1). We assume that the interse
tion of D with the real
]t ,t [
m M
axis is an interval .
λ(t) µ(t) D f (t,φ(t),0)
X
H4 Letusdenote and fortheeigenvaluesoftheja
obianmatrix ,
omputedatpoint .
t
We assume that , for real, the signs of the real and imaginary parts are given by the table below :
t t t
m M
a
(λ(t))
ℜ - 0 +
(µ(t))
ℜ - 0 +
(λ(t))
ℑ - - -
(µ(t))
ℑ + + +
t t t
m M
Then, when in
reases from to , the quasi-steadystate is (cid:28)rst an attra
tivefo
us, then a repulsive
t=a
fo
us, with a Hopf bifur
ation at .
2.1 Input-output fun
tion when there exists a big
anard
X˜(t) X˜(t) φ(t)
2
Inthisse
tion,weassumethatthereexistsabig
anard i.e. asolutionofequation(1)su
hthat ≃
t S ]t ,t [
m M
for alXl˜ in the -interiorof . We now want to study the others solutions of equation (1) by
omparison
with .
X
Themaintoolforthatisasequen
eof
hangeofunknowm: (cid:28)rst,weperformatranslationon ,depending
t
on to put the big
anard on the axis: X = X˜(t) + Y
It gives the system
εY˙ = g(t,Y,ε) g(t,Y,ε)=f(t,X˜(t)+Y,ε) f(t,X˜(t),ε)
with −
ε ε arg(ε) ] δ,δ[
1In
lassi
alterms,weassumethat leavesinasmall
omplexse
tor: | |boundedand ∈ − .
ε
2Without nonstandard terminology,abig
anardisasolutionofequation (1)depending ontheparameter su
hthat
∀t∈]tm,tM[,ε>l0i,mε→0X˜(t,ε)=φ(t)
2
D f(t,φ(t),0)
X
Thematrix hastwo
omplex
onjugatedistin
teigenvalues(seehypothesisH4),thenthereexists
P(t)
a linear transformation whi
h transforms the ja
obian matrix in a
anoni
al form. We de(cid:28)ne the
hange
of unknown
Y =P(t)Z
The new system, has the following form (we wrote only the interesting terms):
α(t) ω(t)
εZ˙ = h(t,Z,ε) h(t,Z,ε)= − Z+O(ε)Z +O(Z2) λ(t)=α(t) iω(t)
with ω(t) α(t) , −
(cid:18) (cid:19)
The next
hange is given by the polar
oordinates:
rcosθ
Z =
rsinθ
(cid:18) (cid:19)
εr˙ = r(α(t)+O(ε)+O(r))
εθ˙ = ω(t)+O(ε)+O(r)
(cid:26)
3
The last one is an exponential mi
ros
ope : ρ
r = exp
ε
(cid:16) (cid:17)
ρ˙ = α(t)+O(ε)+eρεk1(r,θ,ε)
(cid:26) εθ˙ = ω(t)+O(ε)+eρεk2(r,θ,ε) (2)
ρ r
While is non positive and non in(cid:28)nitesimal, is exponentially small and the equation (2) gives a good
ρ ρ˙ = α ρ
approximation of with . When be
omes in(cid:28)nitesimal, with a more subtle argument (see [1℄) using
r
di(cid:27)erential inequations, we
an prove that be
omes non in(cid:28)nitesimal. This gives the proposition below:
Proposition 1 Let us assXu˜m(te)hypothesis H1 to H4 (Hopf bifur
ation) f]otr e,qtua[tion (1). HXow(te)ver, we assume
4 m M
that there exists a
anard going along the slow
urve at least on . Then if goes along the
]t ,t [ [t ,t ] ]t ,t [
5 e s e s m M
slow
urve exa
tly on with ⊂ , then
ts
(λ(τ))dτ = 0
ℜ
Zte
t t ts (λ(τ))dτ = 0
The input-output relation (between e and s) is de(cid:28)ned by te ℜ . It is des
ribed by its graph
(see (cid:28)gure 1). In this
ase, this relation is a fun
tion. R
Figure 1: The input-output relation for equation (3) when there exists a big
anard.
ε ε=0
43AAllsotlhuetiporne
X˜ee(dt,inε)ggtoraenssafolornmgatthioensslowwer
eurrevgeualatrlewaistthornes]pt1e,
tt2t[oif . Thislastoneissingularat .
t ]t1,t2[, lim X˜(t,ε)=φ(t)
∀ ∈ ε>0,ε→0
5A solution X˜(t,ε) goes along the slow
urve exa
tly on ]te,ts[ if it goes along the slow
urve at least on ]te,ts[, and if the
]te,ts[
interval ismaximalforthisproperty.
3
2.2 The bump and the anti-bump
t t
t t
In this paragraph, be
omes
omplex, in the domain D . We assume that for all in D , the two eigenvalues
λ(t) µ(t)
and are distin
t. It is a ne
essary
ondition to apply Callot's theory of reliefs.
R R
λ µ
We de(cid:28)ne the reliefs and by:
t
F (t) = λ(t)dt R (t)= (F (t))
λ λ λ
, ℜ
Za
t
F (t) = µ(t)dt R (t)= (F (t))
µ µ µ
, ℜ
Za
λ(t) = µ(t) F (t) = F (t) R (t) = R (t) R R
λ µ λ µ λ µ
It is easy to see that , and , then . The two fun
tions and
R(t)

oin
ide on the real axis. We will denote .
γ :s [0,1] R d R (γ(s))<0
De(cid:28)nition 1 We say that a path ∈ 7→Dt goes down the relief λ if and only if ds λ for
s [0,1]
all in .
t (t ,φ(t ),0) t
e e e t e
De(cid:28)nition 2 Let us give a point su
h that ∈ D. We say that D is a domain below if and
t S t t
t t e
only if for all in the -interior of D , there exist two paths in D , from to , the (cid:28)rst one goes down the
R R
λ µ
relief and the se
ond one down .
t X(t)
t e
Theorem 2 (Callot) Let us assume that D is a domain below . A solution of equation (1) with
X(t ) φ(t ) S
e e t
an initial
ondition in(cid:28)nitesimally
lose to is de(cid:28)ned at least on the -interior of D where it is
φ(t)
in(cid:28)nitesimally
lose to .
Let us apply this theorem to the followingexample,
hosen as the typi
al example satisfying hypothesis H1
to H4 (Hopf bifur
ation).
εx˙ = tx+y+εc
1
εy˙ = x+ty+εc2 (3)
(cid:26) −
λ=t i µ=t+i R (t)= 1(t i)2 R (t)= 1(t+i)2
Theeigenvaluesare − et . Thelevel
urvesofthetworeliefs λ 2 − and µ 2
are drawn on (cid:28)gure 2.
t
e
Figure 2: The level
urves of the two reliefs of equation (3), and a domain below
t = i
1
Generi
ally there is no surstability at point (see [3℄ for the de(cid:28)nition of surstability). Consequently,
we have the following results for all equations su
h that the reliefs are on the same type as those of (cid:28)gure 2:
t λ(t )=0
c 2 c
De(cid:28)nition 3 Let us give a point where the eigenvalue vanishes: . The value of the relief at point
t R t a R (t )
c λ 3 ∗ λ ∗
is a
riti
al value of the relief . The bump is the real number bigger than , minimal su
h that
t a R (t )
∗∗ λ ∗∗
is a
riti
al value. The anti-bump is the real number smaller than , maximal su
h that is a
riti
al
value.
1Idonotknowtheexa
tgeneri
hypothesis. Wehaveto
ombinethe
onstraintsgivenbythesurstabilitytheoryof[3℄andthe
fa
tthat theequation (1)isreal
2Insome
ases,itispossiblethat tc isin(cid:28)nite. ForεX˙ =„ sicnostt csoinstt «X+O(X2)+O(ε)wehave tc=+i∞.
−
3Thename"bump"isatranslationofthefren
h name"butée"
4
t =1 t = 1
∗ ∗∗
For equation (3), the bump is and the anti-bump − .
X =φ(t) ]t ,t [
e s
Theorem 3 A traje
tory of equation (1)
an go along the slow
urve exa
tly on if and only if
one of the following is veri(cid:28)ed:
te <t∗∗ ts =t∗
and
t =t t >t
e ∗∗ s ∗
and
t∗∗ <te <a a<ts <t∗ R(te)=R(ts)
and and
This theorem is illustrated by the graph of the input-output relation, drawn on (cid:28)gure 3.
Figure 3: The input-output relation for equation (3)
3 Delayed Hopf bifur
ation followed by a fo
us-node bifur
ation
The studied equation is εX˙ =f(t,X,ε)
(4)
f 2
where is analyti
on a domain D of C×C ×C, and satis(cid:28)es the following hypothesis:
Hypothesis and notations
f
HFN1 The analyti
fun
tion takes real values when the arguments are real.
ε
HFN2 The parameter is real, positive, in(cid:28)nitesimal.
φ f(t,φ(t),0) = 0
t
HFN3 There exists an analyti
fun
tion , de(cid:28)ned on a
omplex domain D su
h that . The
X =φ(t)
t

urve is
alledthe slow
urve ofequation1. Weassumethatthe interse
tionofD with thereal
]t ,t [
m M
axis is an interval .
λ(t) µ(t) D f (t,φ(t),0)
X
HFN4 Letusdenote and fortheeigenvaluesoftheja
obianmatrix ,
omputedatpoint .
t
We assume that , for real, the signs of the real and imaginary parts are given by the table below :
t t t
m M
a b
(λ(t))
ℜ - 0 + + +
(µ(t))
ℜ - 0 + + +
(λ(t))
ℑ - - - 0 0
(µ(t))
ℑ + + + 0 0
t ]t ,t [
m M
Then, when in
reases on the real interval , we have su

esively an attra
tive fo
us, a Hopf
t = a t = b
bifur
ation at , a repulsive fo
us, a fo
us-node bifur
ation at and a repulsive node. At point
t=b λ(t)=µ(t) b
, the two eigenvalues
oin
ide. We assumethat only at point . A
tually, in the
omplex
plane, the two eigenvalues are the two determinations of a multiform fun
tion de(cid:28)ned on a Riemann
b
surfa
e with a square root singularity at point .
√
However, there is a symmetry: if the fun
tion is de(cid:28)ned with a
ut-o(cid:27) on the positive real axis, it
√s= √s
satis(cid:28)es − and we then have
µ(t) = λ(t)
5
HFN5 For the same reason, the two reliefs
t t
R (t)= λ(t)dt R (t)= µ(t)dt
λ µ
ℜ and ℜ
(cid:18)Za (cid:19) (cid:18)Za (cid:19)
t=b
arethetwodeterminationsofamultiformfun
tionwithasquarerootsingularityatpoint . However,
√
there is a symmetry: if the fun
tion is de(cid:28)ned with a
ut-o(cid:27) on the positive real axis, it satis(cid:28)es
√s = √s R (t) = R (t) [b,+ [ t > b
µ λ
− and we have then: ex
ept on the
ut-o(cid:27) half line ∞. For real ,
λ(t) <µ(t) R
λ
we
hoose determinations of square root su
h that . We assume that has a unique
riti
al
R R (b)<R
c λ c
point with
riti
al value . We assume that . An example is given and studied in paragraph
3.2.1.
3.1 Input-output fun
tion when there exists a big
anard
X˜(t) X˜(t) φ(t)
We assume now that there exists a big
anard i.e. a solution of equation (1) su
h that ≃ for
t S ]t ,t [
m M
all in the -interior of . The study below is similar to paragraph 2.1. The added di(cid:30)
ulty is the
b

oin
iden
e of the two eigenvalues aXt p=oiXn˜t(t)w+hZi
h do not allow to diagonalize the linear pXart=. 0
The (cid:28)rst
hange of unknown is whi
h moves the big
anard on the axis :
εZ˙ = A(t)Z +O(ε)Z +O(Z2) A(t)=D f(t,φ(t),0)
X
,
α(t) β(t)
A(t)
Letusdenote γ(t) δ(t) the
oe(cid:30)
ientsofthematrix . Asin paragraph2.1, the
hangeofunknowns
(cid:18) (cid:19)
rcosθ ρ
Z = r = exp
rsinθ , ε
(cid:18) (cid:19) (cid:16) (cid:17)
gives the new system:
ρ˙ = α(t)cos2θ+(β(t)+γ(t))cosθsinθ+δ(t)sin2θ+O(ε)+eρεk1(r,θ,ε)
(cid:26) εθ˙ = γ(t)cos2θ+(δ(t)−α(t))cosθsinθ−β(t)sin2θ+O(ε)+eερk2(r,θ,ε) (5)
ρ r
Fornonpositive (morepre
isely,forin(cid:28)nitesimal ), these
ondequationisaslow-fastequation. Itsslow
urve
is given by
δ(t) α(t) α(t)2 2α(t)δ(t)+δ(t)2+4β(t)γ(t)
θ = arctan − ± −
 q 2β(t) 
 
λ µ
It has two bran
hes when and are reals, one is attra
tive, the other is repulsive: see (cid:28)gure 4.
θ r
When goes along a bran
h of the slow
urve, (and when is in(cid:28)nitesimal), an easy
omputation shows
ρ˙ λ µ
that is in(cid:28)nitely
lose to one of the eigenvalues or . The repulsive bran
h
orresponds to the smallest
t < b θ
eigenvalue (whi
h is real positive). When , the angle moves in(cid:28)nitely fast, and an averaging pro
edure
ρ
is needed to evaluate the variation of : θ1+2π ρ˙dθ
ρ˙ = θ1 θ˙
h i Rθ1+2π 1dθ
θ1 θ˙
S R t<b ρ<0
An easy
omputation shows now that, in the -interiorof the domain , , we have
α(t)+δ(t)
ρ˙ = (λ(t))= (µ(t))
h i ≃ 2 ℜ ℜ
(t,θ) ρ
Let us give an initial
ondition between the two bran
hes of the slow
urve and negative non
t = 0.8 θ = 0 ρ = 0.03 t (t,θ(t))
in(cid:28)nitesimal (in the example, we
an take , , − ). For in
reasing , the
urve goes
ρ t
along the attra
tive bran
h of the slow
urve, while believes negative non in(cid:28)nitesimal. For de
reasing , the
θ ρ
solutiongoesalongtherepulsivebran
h,then movesin(cid:28)nitelyfastwhile believesnegativenonin(cid:28)nitesimal.
ρ(t)
Consequently, we know the variation of (see (cid:28)gure 4). As in paragraph 2.1, a more subtle argument is
ρ r
needed to provethat when be
omes in(cid:28)nitesimal, the variable be
omes non in(cid:28)nitesimal and the traje
tory
X
leaves the neighborhood of the slow
urve.
ρ(t)
From this study, all the behaviours of are known, depending on the initial
ondition. They are drawn
on (cid:28)gure 5.
6
t 1
0.002X˙ = X (θ,ρ)
Figure 4: One of the traje
tories of system t 0.3 t drawn with the variables . The
(cid:18) − (cid:19)
slow
urve is also drawn
ρ(t)
Figure 5: The possible behaviours of .
Proposition 4 LetX˜u(st)give an equation of type (6) with hypothesis HFN1]tto ,HtFN[5. Assume also Xtha(tt)there
m M
exists a big
anard going along the slow
urve on the whole interval . If a traje
tory goes
]t ,t [ [t ,t ] ]t ,t [
e s e s m M
along the slow
urve exa
tly on an interval with ⊂ , then
ts ts
(λ(τ))dτ 0 (µ(τ))dτ
ℜ ≤ ≤ ℜ
Zte Zte
Conversely, if the inequalities above are satis(cid:28)ed, there exists a traje
tory going along the slow
urve exa
tly
]t ,t [
e s
on .
The input-output relation is des
ribed by its graph, drawn on (cid:28)gure 6.
r θ
We
ould give more pre
ise results if we
onsider the two variables and for the input-output relation.
(t ,t )
e s
Indeed,whenthepoint isinthe interiorofthegraphoftheinput-output relation,weknowthat,attime
θ
of output, is goingalong the attra
tiveslow
urvewhi
h
orrespondsto the unique fast traje
torytangent to
µ
the eigenspa
e of the biggest eienvalue .
3.2 The fo
us-node bifur
ation is a bump
Hereisthemainpartofthisarti
le. Today,Iamnotabletoprovetheexpe
tingresults,butIhavepropositions
in this dire
tion. To explain the problem, I will give
onje
tures.
t t t
Let us de(cid:28)ne the anti-bump ∗∗ and the two bumps ∗λ and ∗µ as in de(cid:28)nition 3:
R (t ) = R (t ) = R (t ) = R (t ) = R (t ) = R (t )
λ c µ c λ ∗∗ µ ∗∗ λ ∗λ µ ∗µ
7
Figure 6: The input-output relation for equation (6) when there exists a big
anard.
t < a < t = t b t < a < b < t < t
. We have ∗∗ ∗µ ∗λ ≤ or ∗∗ ∗µ ∗λ. In the (cid:28)rst
ase, the bump is before the
fo
us node bifur
ation, and the study of paragraph 2.2 is available. The interesting
ase is the se
ond, where
the
omputed bump is after the fo
us node bifur
ation, this
ase is assumed with hypothesis HFN5.
Conje
ture 5 With hypothesis HFN1 to HFN5, the following proposition is generi
ally wrong:
]t ,a[
∗∗
If a traje
tory of (4) goes along the slow
urve at least on , then it goes until the slow
urve at least
[t ,t ]
on ∗∗ ∗µ .
Toworkonthis
onje
ture,wewillstudyanexamplewhi
his, insomesense, anormalformoftheproblem:
t
the slow
urve is moved on the -axis and the fast ve
tor (cid:28)eld is linearized. The example is
ε3x˙ = tx+y+ε3c
1
ε3y˙ = (t b)x+ty+ε3c2 (6)
(cid:26) −
Proposition 6 A numeri
al simulation of equation (6) gave the (cid:28)gure 7. It
on(cid:28)rms
onje
ture 5.
X X b
+
Figure7: Traje
tories − and : the(cid:28)rstgoesalongthehorizontalaxisfrom−∞to ,whereitjumpsoutside
+ b
the neighborhoodofthe horizontalaxis; the se
ondonegoesalongthe horizontalaxisfrom ∞to − where it
b= 0.3 c =0 c = 1 ε3 = 0.002
1 2
has big os
illations. The parameters are , , − , , the traje
tory is
omputed with
0.0001
a RK4 method, with step . Other methods and other steps were tried, and the results are always very
similar.
This proposition gives a good argument for the next
onje
ture, more pre
ise than the (cid:28)rst one:
t t < a
Conje
ture 7 If a traje
tory of system (4) goes along the slow
urve in a neighborhood of a real with
R(t)>R(b) b
and , then it does not go along the slow
urve after the fo
us-node bifur
ation point .
8
R(t )>
∗∗
So,generi
ally,theinput-output relationofequation(4)hasagraphsimilartothegraphof(cid:28)gure3;if
R(b) t b t t R(t )=R(b)
,wehavetorepla
e ∗ by et ∗∗ par ∗b∗ where ∗b∗ . ThedelayoftheHopfbifur
ationisstopped
either by the bump (as in
ase of a Hopf bifur
ation alone) either by the fo
us-node bifur
ation.
Proposition 8 If the
onje
ture 7 is true for one traje
tory, then it is true for all of them.
X˜ X˜
Proof Assume that equation (4) has a solution whi
h does not verify
onje
ture 7. Then, goes along
]t ,t [ t < t < a < b < t
the slow
urve on an interval 1 2 with 1 ∗b∗ 2. If the problem is
onsidered on a restri
ted
]t ,t [
1 2
interval , the equationthas a big
anard, and we
an applby the proposition 4. Then all traje
tories goin(cid:4)g
along the slow
urve before ∗b∗ goes along the slow
urve until , and even a little more.
ε ε3
In this arti
le, we will now study only equation (6). We
hanged into only to avoid fra
tionnary
ε
exponents. The analyti
stru
ture withrespe
tto isobviouslymodi(cid:28)ed, but does notmatterforourpurpose.
To study the phase portrait of equation (4) or (6), two traje
tories are very important. They are
alled
X
+
distinguished traje
tories by JL.Callot and they are very
lassi
al. The (cid:28)rst one, denoted goes along the
t t X t t
M m
slow
urve for near . Similarly, − goes along the slow
urve for near . These two traje
tories are
t = t =+
m M
Feni
hel'smanifolds,they areuniquewhen −∞and ∞. Fortheparti
ularequation(6),thesetwo
X X
+
traje
tories are drawn on (cid:28)gure 7. We have for this example a ni
e fa
t: − and have an expli
it formula,
usingtheAiryfun
tion(inanappendix(se
tion4), wegive
lassi
alneeded resultsonAiryfun
tions andAiry
equation). X+(t) = (cid:18) xy++((tt)) (cid:19) = −e21εt23M(t)Zt+∞e−12τε32M−1(τ)dτ(cid:18) cc12 (cid:19) (7)
X−(t) = xy−((tt)) = e21εt23M(t) t e−12τε32M−1(τ)dτ cc12 (8)
(cid:18) − (cid:19) Z−∞ (cid:18) (cid:19)
π A jt b A j2t b i
M(t) = ε−2 ε−2 det(M(t)) =
where rε (cid:18) εjA(cid:0)′ jtε−2(cid:1)b εj2A(cid:0)′ j2tε−(cid:1)2b (cid:19) with 2 (9)
(cid:0) (cid:1) (cid:0) (cid:1) C t−32e32|t|32
All the integrals are
onvergent be
ause the Airy fun
tion is bounded at in(cid:28)nity by | | .
3.2.1 The relief
Inthisparagraph,wewanttoexplorethemethodsusedinparagraph2.2whenthereisafo
us-nodebifur
ation.
We also
he
k the hypothesis HFN1 to HFN5.
φ(t,X,0)=0 = 2
HypothesisHFN1 to HFN3areobviouswith theslow
urve and thedomainD C×C ×C.
t 1
J(t)=
The
omputation of the eigenvalues of the ja
obian matrix t b t gives
(cid:18) − (cid:19)
1 1
λ(t)=t (t b)2 µ(t)=t+(t b)2
− − −
The determinationof thesquarerootisneeded to allowtheformulaabove. In all this paragraph,we
hoose
a
ut-o(cid:27) on the positive real axis: (reiθ)21 = √r ei2θ θ [0,2π[
∈
3
()2
For the fun
tito12n= ,tw12e
hoose the same
ut-oλ(cid:27). µ
The relation − will be useful. Then, and are the two determinations of a multiform fun
tion.
[b,+ [ µ(t)=λ(t)
The
ut-o(cid:27) is the semi-axis ∞, and .
a=0
For and
b> 1,
4 (10)
the hypothesis HFN4 is easy to
he
k.
The two asso
iated reliefs are given by
Fλ(t)= 12t2− 32(t−b)32 − 23ib23 Fµ(t)= 21t2+ 23(t−b)23 + 32ib23
R (t)= (F (t)) R (t)= (F (t))
λ λ µ µ
ℜ ℜ
9
R b=0.3
λ
Figure 8: Level
urves of relief for , and path used in paragraph3.2.4.
R + t=
λ
Let us
omment (cid:28)gure 8: the value of is ∞ at both ends of the real axis. If a path goes from −∞
t = +
to ∞, it has to go down at least until the mountain pass, whi
h is the unique
riti
al point of the relief
given by
t = 1 +i b 1 = 0.500+0.224 i
c 2 − 4
p
The value of the relief at this
riti
al point is
R =R (t ) = 1b 1 = 0.067
c λ c 2 − 12
R (t)=R t t
λ c e s
We solve now on the real axis the equation . The solution are and given by
b> 1 + 1√3 , t = b 1 t = b 1
if 2 6 e − − 6 s − 6
t = ...
 if b< 12 + 61√3 , te = −pb− 61 tss21 =p ...
(cid:26)
p
...  t 4
The symbols in the formula above are the solutions of a polynom in of degree . The exa
t expression is
b=0.3
not needed. For , we have
t = 0.365 t =0.346 t =0.525
e s1 s2
−
t arg(t )=2π t
s1 s1 s2
The value is on the sheet right to the
ut-o(cid:27): . Besides, is on the sheet left to the
ut-o(cid:27):
arg(t ) = 0 t t
s2 s1 s2
. When we look on the polynom whi
h has and as roots, we
an prove that the hypothesis
HFN5 is satis(cid:28)ed for 1 < b < 1 + 1√3
4 2 6 (11)
3.2.2 Callot's domains
To study the
anards of equation (6), we introdu
e two spe
ial solutions,
alled distinguished solutions by J.L.
X = (x ,y ) X (+ ) = 0 X = (x ,y )
+ + + 1 +
Callot: X ( )=0 has an asymptoti

ondition ∞ and − − − Xhas an asymptoti
+ +

ondition − −∞ . They are unique. In this paragraph we build a domain D where is in(cid:28)nitesimal
(it
orresponds in the
omplex plane to the expression "going along a real interval"). In allmost all situations,
the builded domain is the maximal domain with this property.
1Herethethingsare easierthaninthegeneral
asebe
ause thedomainDt
ontains thewholereal axis. In general
ase,there
isnouni
ityofthedistinguishedsolution,butthedi(cid:27)eren
e remainsexponentially smallerthanthe
omputedquantities.
10