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