Approximate invariant manifolds up to exponentially small terms
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English

Approximate invariant manifolds up to exponentially small terms

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Niveau: Supérieur, Master, Bac+4
Approximate invariant manifolds up to exponentially small terms Gerard Iooss 1 Eric Lombardi 2 1I.U.F., Universite de Nice, Labo J.A.Dieudonne, Parc Valrose, 06108 Nice, France 2Universite Paul Sabatier, Institut de Mathematiques, 31062 Toulouse, France , Abstract This paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspaces E0 (dim m0), E1 (dim m1). Under light diophantine conditions on the linear part, we prove that there is a polynomial change of coordinate in E1 allowing to eliminate, in the E1 component of the vector field, all terms depending only on the coordinate u0 ? E0, up to an ex- ponentially small remainder. This main result enables to prove the existence of analytic center manifolds up to exponentially small terms and extends to infinite dimensional vector fields. In the elliptic case, our results also proves, with very light assumptions on the linear part in E1, that for initial data very close to a certain analytic manifold, the solution stays very close to this manifold for a very long time, which means that the modes in E1 stay very small. Keywords: analytic vector fields; normal forms; exponentially small remainders; center manifolds AMS: 34M45; 34G20 1 Introduction Let us consider an analytic vector field in the neighborhood of an equilibrium which we take at the origin.

  • vector field

  • subspaces e0

  • all terms

  • analytic vector

  • exponentially small

  • any analytic

  • manifold

  • center manifold


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Nombre de lectures 17
Langue English
Appr
oximate invariant manifolds up to exponentially small terms G´rard Iooss1Eric Lombardi2 e
1so,e6001aPcraVrlance8Nice,Fr.I.Uinev.FU,de´eitrsab,LceNieiD.A.Jo,e´nnodu 2UniluaPabaSsreve´titutieMtdertins,I,s13qieumetata´hFranuse,oulo062Tec gerard.iooss@unice.fr, lombardi@math.univ-toulouse.fr
1
Abstract
This paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspacesE0 (dimm0), E1(dimm1)Under light diophantine conditions on the linear part, we prove that there is a polynomial change of coordinate inE1allowing to eliminate, in theE1component of the vector field, all terms depending only on the coordinateu0E0, up to an ex-ponentially small remainder. This main result enables to prove the existence of analytic center manifolds up to exponentially small terms and extends to infinite dimensional vector fields. In the elliptic case, our results also proves, with very light assumptions on the linear part inE1,that for initial data very close to a certain analytic manifold, the solution stays very close to this manifold for a very long time, which means that the modes inE1stay very small. Keywords: analytic vector fields; normal forms; exponentially small remainders; center manifolds
AMS: 34M45; 34G20
Introduction
Let us consider an analytic vector field in the neighborhood of an equilibrium which we take at the origin. A natural idea is to try to uncouple a subset of coordinates from the other ones, by using a change of variables. This isusedinparticularsincePoincare´andDulac,andthisisoneofthemain tool in the search of invariant manifolds of vector fields. Eliminating most of components of the vector field, expecting to only keep the relevant ones for the dynamics, is precisely the idea of center manifold reduction, which
1
is widely used in many physical systems, to simplify the study of the dy-namics. However this reduction is only valid when we want to eliminate the hyperbolic part of the vector field and it has the defect to kill the analyticity after the reduction process. For systems fully elliptic near the origin, it may be expected to use a change of variables to uncouple all oscillatory modes. If this were possible, and if the initial data does not excite some modes, these ones would not be awaken for all times. Unfortunately, this is not possible in general, even though for hamiltonian systems, with suitable non resonant eigenvalues of the linearized system, it is nearly the case (Arnold diffusion between invariant tori corresponding to the ”normal form” system with uncoupled modes). In the present work, we consider systems for which the linearized system is split in two invariant subspacesE0(dimm0), E1(dimm1)With light assumptions on the linear part, our main result is that there is a polynomial change of coordinate inE1allowing to eliminate, in theE1component of the vector field, all terms depending only on the coordinateu0E0, up to an exponentially small remainder (see Theorem 1). The proof of this theorem is based on a Gevrey estimate of the divergence of the remainder, which can be exponentially small by an optimal choice of the degree of the polynomial change of coordinates. Gevrey estimates of the divergence of remainders, to get exponentially small upper bounds after an optimal choice of the order, were already used in the theory of normal forms for Hamiltonian systems in action-angle coor-dinates [2], [3], [14] following the pioneering work of Nekhoroshev [11, 12]. A similar result of exponential smallness of the remainder was also obtained by Giorgilli and Posilicano in [4] for areversible systemwith a linear part composed of harmonic oscillators. For an extension of the result of normal forms with an exponentially remainder to any analytic vector fields with semi simple linear part see [7]. Direct normalization up to exponentially small terms is not available for vector fields studied in this paper sinceL1is not assumed to be diagonaliz-able. However we can eliminate from theE1component of the vector field all terms depending only on the coordinateu0E0, up to an exponentially small remainder. A first application of this result is when the linear part inE1is hyper-bolic, while the linear part inE0has all its eigenvalues on the imaginary axis. It is well known that the center manifold reduction applies for small bounded solutions [8], which then lie on a manifold of same dimension asE0It is also well known that this manifold is in general not analytic [13], [20], [1], [16]. Our result allows to obtain a center manifold which is the graph of
2
a function sum of a polynomial of degreep=O(δb) and an exponentially small function of orderO(ecδb) whereδsize of the ball where weis the study the solutions, andcandbare positive numbers (see Theorem 5). It results in particular that the loss of analyticity is located in exponentially small terms. This result extends in infinite dimensional cases, then appli-cable in particular for a large class of PDE’s. So combining, this result on center manifolds with the normal form theorem with exponentially small re-mainder [7] for theE0component (L0is diagonalizable), we can transform (1) into a new system with a ”simplified” analytic leading part, perturbed by exponentially small terms. Such a transformation can be very useful when dealing with exponentially small phenomena (see [10]). Another application, important in particular for engineering systems, is when the two linear subsystems inE0andE1have their eigenvalues on the imaginary axis. In particular, this situation happens for non linear vi-brations of structures. Our result gives a sort of justification of a popular elimination process made in a formal way (see for example [9], [15], [17]), which allows to roughly state that for a class of initial data which do not excite in some sense the high frequencies (corresponding toE1), then these ones are not awaken for all times....Our results prove, with very light dio-phantine assumptions (4) on the linear part inE1,that for initial data very close to a certain analytic manifold, the solution stays very close to this manifold for a very long time, which means that the modes inE1stay very small (see theorem 8). This type of result is related to Arnold’s diffusion for Hamiltonian systems (see a related result in [5]), while it should be noticed that we do not assume our system to be Hamiltonian, our assumptions on the eigenvalues being much lighter that usually done on such systems. In particular the linear part inE1is not assumed to be diagonalizable. Finally, notice the particular case studied in the same spirit by Groves and Schneider [6], for whichE0is 2-dimensional and corresponds to a double eigenvalue in 0, whileE1 In this example there iscorresponds to eigenvalues all imaginary. no need of the diophantine condition (4), butE1is infinite dimensional and theresultweobtainhereneedstobeadapted.In[18],Touz´eandAmabili consider the damped case with an external periodic forcing. They assume that high frequency modes lie at a growing distance from the imaginary axis. Our method might be used in such a case, to rigorously prove that the high frequency modes do not awake astgoes to infinity, provided certain non resonance condition between the forcing frequency and natural frequencies are realized, and provided the initial data is taken on a certain manifold in the spirit of Theorem 8.
3
SplittingThm
2
Main results
We gather in this section the main theorems proved in this paper. Our main theorem is the following
Theorem 1Consider the following system inRm(resp.Cm)
ddtuu+R(u),(1) =L whereu(t)Rm(resp.Cm),Lis a linear operator, andRis analytic in a neighborhood of the origin, such that R(u) =XRk[u(k)],(2)
2k
whereRkis ak- linear symmetric map on(Rm)k(resp.(Cm)k) satisfying ||Rk[u1, u2,  , uk]|| ≤cρk||u1||    ||uk||,(3) for a certain radius of convergenceρ >0(here[u(k)]means thek- uple of vectors[u, u,  , u]). Assume that the linear operatorLis the direct sum of two linear operatorsL0onE0(dimm0),andL1onE1(dimm1),such thatL0is diagonalizable with eigenvaluesλ0(1),  , λ(m00)and that there exist constantsγ >0, τ0such that |hα, λ(0)i −λj(1) || ≥γα|τ(4) holds for anyαNm0\{0},and any eigenvalueλj(1)ofL1Then there exists a polynomialΦ:E0E1of degreep=O(δb)such that the change of variables inE1 u1=v1+Φ(u0) (5) transforms the system (1) into the following system inE0×E1: ddut0L0u0+R(0)(u0, v1),(6) = dv1L1v1+R(1)(u0, v1) +ρ(u0), = dt in whichR(0),R(1),ρanalytic in their arguments, and whereare R(0)(u0, u1) =P0R(u0+v1+Φ(u0)), 4
basicSyst
ExpR
AnalyticR
diophCond
Change Var
newSyst
perturbed vector field
P0being the projection onE0which commutes withL,and R(1)(u0, v1) =O(||v1||(||u0||+||v1||)),(7) sup||ρ(u0)|| ≤M eδb,(8) ||u0||≤δ withM, w >0depending only onτ, m0, c, ρ,L1and b= 1 1 +ντ whereνis the maximal index (size of Jordan blocks) of eigenvalues ofL1Remark 2Notice that the constantsMandwdo not depend on the di-mensionm1of the subspaceE1ifL1is a priori in Jordan formThis allows to consider systems with large (even infinite) dimensions. Remark 3Since all the norms are equivalent onRm(resp.Cm), (7),(8) remains true for any norm onRm(resp.Cm). A change of norm simply change the values ofMandw. So, estimates (7),(8) remain true under linear change of coordinates up to a change of values ofMandw. Hence without loss of generality we can assume that the complexified space ofE0 andE1, still denoted byE0andE1read respectivelyE0=Cm0×(0,  ,0) { } m1times andE1= (0,  ,0)×Cm1and that in the canonical basis ofCm, L0is { } m0times diagonal andL1is under Jordan normal form. We deduce from the above theorem a corollary which deals with vector fields depending on parameters.
Corollary 4Consider the following system inRm(resp.Cm)
ddut=Lu+R(u, ),(9) whereu(t)Rm(resp.Cm),Lis a linear operator, andRis analytic in a neighborhood of the origin inRm×Rq(resp.Cm×Rq)and such that R(0, ) = 0, DuR(0,0) = 0(10) Assuming the same hypothesis onLas in Theorem 1, and that0is not eigenvalue ofL1,then there exists a polynomialΦ:E0×RqE1of degree p=O(δb)such that the change of variables inE1 u1=v1+Φ(u0, )
5
EqRun Eqrho
perturbed syst
0staysSolu
transforms the system (1) into the following system inE0×E1:
du0 =L0u0+R(0)(u0, v1, ), dt dv1 dt=L1v1+R(1)(u0, v1, ) +ρ(u0, ), in whichR(0),R(1),ρare analytic in their arguments, and where
R(0)(u0, u1, ) =P0R(u0+v1+Φ(u0, ), ), P0being the projection onE0which commutes withL,and R(1)(u0, v1, ) =O(||v1||(||u0||+||v1||+||||)), sup||ρ(u0, )||M e, δb ||u0||+||||≤δ with >M, w0depending only onτ, m0, c, ρ,L1andbis as in Theorem 1. Another application of theorem 1, is the existence of analytic center manifolds up to exponentially small term. More precisely, consider the case when the spectrum ofL0iR,andL1 eigenvalues ofis hyperbolic, i.e. the L1lie at a distanceγ > in finite dimension Then0 from the imaginary axis. we have the following
Theorem 5 (Center manifold analytic up to exp. small terms) Consider the analytic system (1) inRmand assume thatL0is diagonalizable with all its eigenvalues on the imaginary axis, and assume thatL1has its eigenvalues at least at a distanceγ >0from the imaginary axis. Then for anyk2,there exists a polynomialΦ:E0E1of degree O(1δ),withΦ(0) = 0, DΦ(0) = 0,a neighborhoodOof0inRm,and a mapΨ∈ Ck(E0, E1)which isO(eδ)for||u0||E0δand a certain constant C >0,such that the manifold
M0={u0+Φ(u0) +Ψ(u0) ;u0E0} has the following properties.
(a)
(11)
M0is locally invariant, i.e., ifuis a solution of (1) satisfyingu(0)M0∩ Oandu(t)∈ Ofor allt[0, T], thenu(t)∈ M0for all t[0, T]
6
EllipticThm
(b)M0set of bounded solutions of (1) staying incontains the Ofor all tR, i.e., ifuis a solution of (1) satisfyingu(t)∈ Ofor alltR, thenu(0)∈ M0
Remark 6The interest of Theorem 5 is that it implies that the reduced system on the center manifold is analytic, up to exponentially small terms. This property is clearly still true after the polynomial new change of variables which put the reduced system under normal form (the usual one). In con-sidering the analytic part of the reduced vector field, this normal form may be derived up to an optimal degree, as made in [7], sinceL0is diagonaliz-able. This may be helpful when dealing with exponentially small phenomena associated with the original system (1).
Remark 7This theorem is also true in the infinite dimensional case (see Theorem 20 in subsection 4.2)
A last application of theorem 1, important in particular for engineering systems, is when the two linear subsystems inE0andE1have both their eigenvalues on the imaginary axis. More precisely in section 5, we prove
Theorem 8(Elliptic vector fields) Assume that assumptions of Theorem 1 hold, and in addition thatL1 for any Thenhas only imaginary eigenvalues. small initial datau(0)chosen on the manifoldM0={u=u0+Φ(u0);u0E0}the solutionu(t)stays at a distanceO(eδb)toM0fort[0, T],with T=O(δ(b+1ν)),whereb += (1ντ)1andνis the maximal index of eigenvalues ofL1
Remark 9We observe (see 39) that in going up to exponentially small terms in Theorem 1, we win the exponential smallness of||v1(t)||for a long range of time, without more precise assumption onL1. If we assume more specific properties of the system, we may have a longer range of time for the validity of this exponential smallness. First, ifL1is diagonalizable this range of time isO(δ[1+(1+τ)1])
Remark 10Let assume in addition that (1) is areversible systemsuch thatLhas only pairs of simple imaginary eigenvalues, satisfying theγ,τ-homologically diophantine assumption defined in [7]: for everyαNm,|α| ≥ 2 |hα, λi −λj| ≥ |αγ|τwhenhα, λi −λj6= 0,
7
andhα, λi−λj= 0only for the trivial cases2λj+λjλj= 0(non resonance assumption). In such a case, we can use the normal form theorem of [7] which gives a normal form up to an exponentially small term, which improves the final form of Theorem 1 since the coupling between the subsystems inE0 and inE1only appears in exponentially small terms.Takingv1(0) = 0,it is easy to show that||v1(t)||stays exponentially small for an exponentially now long time (analogue to Arnold diffusion).
3 Proof of the main theorem
We first deduce corollary 4 from theorem 1 and then we prove this theorem. Proof of Corollary 4. Let us define ue= (u, )Rm×Rq,
then the system reads
with
du ee e dt=Leu+R(eu),
e e Leu= (Lu,0),R(ue) = (R(u, ),0)
(12)
Then, it is clear that the system (12) satisfies all assumptions of Theorem e e e 1. In particular, the operatorLis the direct sum ofL0andL1defined by e f Leu0= (L0u0,0),foreu0E0=E0×Rq, 0 e f L1ue1= (L1u1,0),foreu1E1=E1× {0}, e e and the eigenvalues ofL1are those ofL1,while the eigenvalues ofL0are those ofL0with 0 still semi-simple, having an additionalq- dimensional eigenspace: (0, ), Rqand the diophantine condition (4) is still satisfied. Hence the Corollary is proved.
Proof of Theorem 1.In the proof below we use several algebraic prop-erties which were proved in [7]. Performing the change of coordinates u=u0+u1+ Φ(u0),we check that (1) is equivalent to (6) close to the origin if and only if
P0R(u0+v1+φ(u0)) =R(0)(u0, v1), DΦ(u0)L0u0L1Φ(u0) =DΦ(u0)R(0)(u0, v1)ρ(u0) +P1Ru0+u1+Φ(u0)R(1)(u0, v1)
8
extendSyst
Then Settingv1= 0 and using (7), we obtain the following basic identity DΦ(u0)L0u0L1Φ(u0) =DΦ(u0)P0R(u0+Φ(u0)) +P1R(u0+Φ(u0))ρ(u0)(13) Let decompose the polynomialΦinto a sum of homogeneous polynomials of increasing degrees Φ(u0) =XΦk[u0(k)] 2kp withk- linear symmetric mapsΦk:(E0)kE1For convenience we denote byΦ1(u0)u0which takes its values inE0(contrary toΦkfork2, which takes its values inE1). Then we have for 2np DΦn[u(0n)]L0u0L1Φn[u(0n)] =Fn[u0(n)],(14)
with Fn[u0(n)] =XP1Rq[Φk1,  ,Φkq] + 2qn k1++kq=n kj1 XDΦ[u0()]P0Rq[Φk1,  ,Φkq]2n12qn+1 k1++kq=n+1 kj1 Equation (14) is of the form
AΦn=Fn with the homological operatorAdefined on the vector space of polynomials Φ:E0E1,by AΦ=DΦ(u0)L0u0L1Φ(u0)(15) We then need to introduce the scalar product in the spaceHof polynomials of a variable inE0, taking values inCm(which could be in the complexified space of the subspaceE1orE0) as done in [7]. Given two polynomialsΦandΦwe define their scalar product by hΦ,ΦiH:=XhΦj,Φji 1jn withΦ= (Φ1,  ,Φn),Φ= (Φ1,  ,Φn),and where for a pair of poly-nomialsP, Q:E0C, hP, Qi=P(X)Q(X)|X=0,
9
basicIdent
homologicEqu
homologic
Eric’s
Lemma
where by definition
P(X) =P
X)
Then the associated euclidian norm is defined by |Φ|2:=hΦ,ΦiHIt is clear that for anyn1,the linear operatorAleaves invariant the subspaceHnof homogeneous polynomials of degreen,and we have the following Lemma proved in Appendix:
Lemma 11The operatorAis invertible in the subspaceHnand there exists a constanta,depending only onγandL1,such that |||A|H1n|||2:= sup|A|Hn1Φ| ≤anτ, |Φ|2=1 whereτ=ντ,andνis the maximal index of the eigenvalues ofL1
This lemma is proved in Appendix A.
Remark 12IfL1is in Jordan form, the constantadepends only onγand ν. IfL1is diagonal thenτ=τanda= 1γ.
Moreover, defining the norm
φn:=|Φ|2n:=1n!|Φ|2,forΦ∈ Hn,
we have the following lemma, proved in [7] (see Lemmas 2.10, 2.11):
Lemma 13 (i) Fork1++kq=n |Rq[Φk1,  ,Φkq]|2cρqφ1  φkq, nk
(ii) for2p,+k=n+ 1,and anyNk∈ Hk |DΦNk|2n2+ (m01)ℓ φ|Nk|2kℓ m0φ|Nk|2kThen, the proof of Theorem 1 is performed in several steps giving re-spectively estimates ofφn,kPΦ(u0)k, andρ0gathered in the following lemmas:
10
Lemphin
LemSigmaphiopt
rhoopt
Lemma 14There existsK >0depending only onc, c01, ρ, m0, asuch that for everynwith1np, φnm0Kn1(n!)1+τ(16) wherec01:=max|||P0|||,|||P1|||. Lemma 15Let us choosepsuch that p=popt:=(2δ1K)b, b=+11τ,(17)
where[] fordenotes the integer part of a number. Then||u0|| ≤δwe have XΦk(u0)2δm01kpoptLemma 16The remainderρsatisfies
ρ(u0) =R1(u0) +R2(u0) +R3(u0) +R4(u0),
with R1(u0) =p+X1qP1Rq1XkpΦk(u0)(q), R2(u0) =XDΦ[u0()]P0Rq1XΦk(u0)(q), 2p p+1q kp R3(u0) =XP1Rq[Φk1(u0),  ,Φkq(u0)], 2qp1kjp k1++kqp+1 R4(u0) =XDΦ[u(0)]P0Rq[Φk1(u0),  ,Φkq(u0)], 2p qp1kjp k1+kqpl+2 and forp=popt, it satisfies sup||ρ(u0)|| ≤M eδb,(18) ||u0||≤δ with >M, w0depending only onτ, m0, c, ρ,L1.
11
estimPhi na _
lowerBounda
Eqrhoa