Approximate invariant manifolds up to exponentially small terms

Gérard Iooss

Description

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

Sujets

• Lombardi
• France 2
• Sabatier
• Vector field
• Manifold
• Center manifold

Informations

• Publié par: larec
• Nombre de lectures: 17
• Langue: English

Extrait

Appr

oximate invariant manifolds up to exponentially small terms G´rard Iooss 1 Eric Lombardi 2 e

1 so,e6001aPcraVrlance8Nice,Fr.I.Uinev.FU,de´eitrsab,LceNieiD.A.Jo,e´nnodu 2 UniluaPabaSsreve´titutieMtdertins,I,s13qieumetata´hFranuse,oulo062Tec [e-mail], [e-mail]

1

Abstract

This paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspacesE 0 (dimm 0 ), E 1 (dimm 1 )Under light diophantine conditions on the linear part, we prove that there is a polynomial change of coordinate inE 1 allowing to eliminate, in theE 1 component of the vector field, all terms depending only on the coordinateu 0 ∈E 0 , 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 inE 1 ,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 inE 1 stay 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 subspacesE 0 (dimm 0 ), E 1 (dimm 1 )With light assumptions on the linear part, our main result is that there is a polynomial change of coordinate inE 1 allowing to eliminate, in theE 1 component of the vector field, all terms depending only on the coordinateu 0 ∈E 0 , 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 sinceL 1 is not assumed to be diagonaliz- able. However we can eliminate from theE 1 component of the vector field all terms depending only on the coordinateu 0 ∈E 0 , up to an exponentially small remainder. A first application of this result is when the linear part inE 1 is hyper- bolic, while the linear part inE 0 has 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 asE 0  It 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(δ −b ) and an exponentially small function of orderO(e −cδ b ) 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 theE 0 component (L 0 is 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 inE 0 andE 1 have 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 toE 1 ), then these ones are not awaken for all times....Our results prove, with very light dio- phantine assumptions (4) on the linear part inE 1 ,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 inE 1 stay 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 inE 1 is not assumed to be diagonalizable. Finally, notice the particular case studied in the same spirit by Groves and Schneider [6], for whichE 0 is 2-dimensional and corresponds to a double eigenvalue in 0, whileE 1  In this example there iscorresponds to eigenvalues all imaginary. no need of the diophantine condition (4), butE 1 is 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 inR m (resp.C m )

d dt u u+R(u),(1) =L whereu(t)∈R m (resp.C m ),Lis a linear operator, andRis analytic in a neighborhood of the origin, such that R(u) = X R k [u (k) ],(2)

2≤k

whereR k is ak- linear symmetric map on(R m ) k (resp.(C m ) k ) satisfying ||R k [u 1 , u 2 ,  , u k ]|| ≤ c ρ k ||u 1 ||    ||u k ||,(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 operatorsL 0 onE 0 (dimm 0 ),andL 1 onE 1 (dimm 1 ),such thatL 0 is diagonalizable with eigenvaluesλ 0( 1 ) ,  , λ ( m 0 0 ) and that there exist constantsγ >0, τ≥0such that |hα, λ (0) i −λ j (1)  | | ≥ γ α| τ (4) holds for anyα∈N m 0 \{0},and any eigenvalueλ j (1) ofL 1  Then there exists a polynomialΦ:E 0 →E 1 of degreep=O(δ −b )such that the change of variables inE 1 u 1 =v 1 +Φ(u 0 ) (5) transforms the system (1) into the following system inE 0 ×E 1 : d d u t 0 L 0 u 0 +R (0) (u 0 , v 1 ),(6) = dv 1 L 1 v 1 +R (1) (u 0 , v 1 ) +ρ(u 0 ), = dt in whichR (0) ,R (1) ,ρanalytic in their arguments, and whereare R (0) (u 0 , u 1 ) =P 0 R(u 0 +v 1 +Φ(u 0 )), 4

basicSyst

ExpR

AnalyticR

diophCond

Change Var

newSyst

perturbed vector field

P 0 being the projection onE 0 which commutes withL,and R (1) (u 0 , v 1 ) =O(||v 1 ||(||u 0 ||+||v 1 ||)),(7) sup||ρ(u 0 )|| ≤M e − δb ,(8) ||u 0 ||≤δ withM, w >0depending only onτ, m 0 , c, ρ,L 1 and b = 1 1 +ντ whereνis the maximal index (size of Jordan blocks) of eigenvalues ofL 1  Remark 2Notice that the constantsMandwdo not depend on the di- mensionm 1 of the subspaceE 1 ifL 1 is a priori in Jordan formThis allows to consider systems with large (even infinite) dimensions. Remark 3Since all the norms are equivalent onR m (resp.C m ), (7),(8) remains true for any norm onR m (resp.C m ). 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 ofE 0 andE 1 , still denoted byE 0 andE 1 read respectivelyE 0 =C m 0 ×(0,  ,0) { } m 1 times andE 1 = (0,  ,0)×C m 1 and that in the canonical basis ofC m , L 0 is { } m 0 times diagonal andL 1 is 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 inR m (resp.C m )

d d u t =Lu+R(u, ),(9) whereu(t)∈R m (resp.C m ),Lis a linear operator, andRis analytic in a neighborhood of the origin inR m ×R q (resp.C m ×R q )and such that R(0, ) = 0, D u R(0,0) = 0(10) Assuming the same hypothesis onLas in Theorem 1, and that0is not eigenvalue ofL 1 ,then there exists a polynomialΦ:E 0 ×R q →E 1 of degree p=O(δ −b )such that the change of variables inE 1 u 1 =v 1 +Φ(u 0 , )

5

EqRun Eqrho

perturbed syst

0staysSolu

transforms the system (1) into the following system inE 0 ×E 1 :

du 0 =L 0 u 0 +R (0) (u 0 , v 1 , ), dt dv 1 dt =L 1 v 1 +R (1) (u 0 , v 1 , ) +ρ(u 0 , ), in whichR (0) ,R (1) ,ρare analytic in their arguments, and where

R (0) (u 0 , u 1 , ) =P 0 R(u 0 +v 1 +Φ(u 0 , ), ), P 0 being the projection onE 0 which commutes withL,and R (1) (u 0 , v 1 , ) =O(||v 1 ||(||u 0 ||+||v 1 ||+||||)), sup||ρ(u 0 , )||M e − , ≤ δb ||u 0 ||+||||≤δ with >M, w0depending only onτ, m 0 , c, ρ,L 1 andbis 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 ofL 0 ⊂iR,andL 1  eigenvalues ofis hyperbolic, i.e. the L 1 lie 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) inR m and assume thatL 0 is diagonalizable with all its eigenvalues on the imaginary axis, and assume thatL 1 has its eigenvalues at least at a distanceγ >0from the imaginary axis. Then for anyk≥2,there exists a polynomialΦ:E 0 →E 1 of degree O(1δ),withΦ(0) = 0, DΦ(0) = 0,a neighborhoodOof0inR m ,and a mapΨ∈ C k (E 0 , E 1 )which isO(e − δ )for||u 0 || E 0 ≤δand a certain constant C >0,such that the manifold

M 0 ={u 0 +Φ(u 0 ) +Ψ(u 0 ) ;u 0 ∈E 0 } has the following properties.

(a)

(11)

M 0 is locally invariant, i.e., ifuis a solution of (1) satisfyingu(0)∈ M 0 ∩ Oandu(t)∈ Ofor allt∈[0, T], thenu(t)∈ M 0 for all t∈[0, T]

6

EllipticThm

(b)M 0 set of bounded solutions of (1) staying incontains the Ofor all t∈R, i.e., ifuis a solution of (1) satisfyingu(t)∈ Ofor allt∈R, thenu(0)∈ M 0 

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], sinceL 0 is 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 inE 0 andE 1 have 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 thatL 1  for any Thenhas only imaginary eigenvalues. small initial datau(0)chosen on the manifoldM ′ 0 ={u=u 0 +Φ(u 0 );u 0 ∈ E 0 }the solutionu(t)stays at a distanceO(e − δb )toM ′ 0 fort∈[0, T],with T=O(δ −(b+1ν) ),whereb += (1ντ) −1 andνis the maximal index of eigenvalues ofL 1 

Remark 9We observe (see 39) that in going up to exponentially small terms in Theorem 1, we win the exponential smallness of||v 1 (t)||for a long range of time, without more precise assumption onL 1 . 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, ifL 1 is 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 everyα∈N m ,|α| ≥ 2 |hα, λi −λ j | ≥  |α γ | τ whenhα, λi −λ j 6= 0,

7

andhα, λi−λ j = 0only for the trivial cases2λ j +λ j −λ j = 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 inE 0 and inE 1 only appears in exponentially small terms.Takingv 1 (0) = 0,it is easy to show that||v 1 (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, )∈R m ×R q ,

then the system reads

with

du  e e 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 ofL 0 andL 1 defined by e f Leu 0 = (L 0 u 0 ,0),foreu 0 ∈E 0 =E 0 ×R q , 0 e f L 1 ue 1 = (L 1 u 1 ,0),foreu 1 ∈E 1 =E 1 × {0}, e e and the eigenvalues ofL 1 are those ofL 1 ,while the eigenvalues ofL 0 are those ofL 0 with 0 still semi-simple, having an additionalq- dimensional eigenspace: (0, ), ∈R q and 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=u 0 +u 1 + Φ(u 0 ),we check that (1) is equivalent to (6) close to the origin if and only if

P 0 R(u 0 +v 1 +φ(u 0 )) =R (0) (u 0 , v 1 ), DΦ(u 0 )L 0 u 0 −L 1 Φ(u 0 ) =−DΦ(u 0 )R (0) (u 0 , v 1 )−ρ(u 0 ) +P 1 R  u 0 +u 1 +Φ(u 0 )  −R (1) (u 0 , v 1 )

8

extendSyst

Then Settingv 1 = 0 and using (7), we obtain the following basic identity DΦ(u 0 )L 0 u 0 −L 1 Φ(u 0 ) =−DΦ(u 0 )P 0 R(u 0 +Φ(u 0 )) +P 1 R(u 0 +Φ(u 0 ))−ρ(u 0 ) (13) Let decompose the polynomialΦinto a sum of homogeneous polynomials of increasing degrees Φ(u 0 ) = X Φ k [u 0 (k) ] 2≤k≤p withk- linear symmetric mapsΦ k :(E 0 ) k →E 1 For convenience we denote byΦ 1 (u 0 )≡u 0 which takes its values inE 0 (contrary toΦ k fork≥2, which takes its values inE 1 ). Then we have for 2≤n≤p DΦ n [u ( 0 n) ]L 0 u 0 −L 1 Φ n [u ( 0 n) ] =F n [u 0 (n) ],(14)

with F n [u 0 (n) ] = X P 1 R q [Φ k 1 ,  ,Φ k q ] + 2≤q≤n k 1 ++k q =n k j ≥1 − X DΦ ℓ [u 0 (ℓ) ]P 0 R q [Φ k 1 ,  ,Φ k q ] 2≤ℓ≤n−12≤q≤n−ℓ+1 k 1 ++k q =n−ℓ+1 k j ≥1 Equation (14) is of the form

AΦ n =F n with the homological operatorAdefined on the vector space of polynomials Φ:E 0 →E 1 ,by AΦ=DΦ(u 0 )L 0 u 0 −L 1 Φ(u 0 )(15) We then need to introduce the scalar product in the spaceHof polynomials of a variable inE 0 , taking values inC m (which could be in the complexified space of the subspaceE 1 orE 0 ) as done in [7]. Given two polynomialsΦandΦ ′ we define their scalar product by hΦ,Φ ′ i H := X hΦ j ,Φ j i ′ 1≤j≤n withΦ= (Φ 1 ,  ,Φ n ),Φ ′ = (Φ ′ 1 ,  ,Φ ′ n ),and where for a pair of poly- nomialsP, Q:E 0 →C, 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,Φi H  It is clear that for anyn≥1,the linear operatorAleaves invariant the subspaceH n of homogeneous polynomials of degreen,and we have the following Lemma proved in Appendix:

Lemma 11The operatorAis invertible in the subspaceH n and there exists a constanta,depending only onγandL 1 ,such that |||A| H −1 n ||| 2 := sup|A| − H n 1 Φ| ≤an τ ′ , |Φ| 2 =1 whereτ ′ =ντ,andνis the maximal index of the eigenvalues ofL 1 

This lemma is proved in Appendix A.

Remark 12IfL 1 is in Jordan form, the constantadepends only onγand ν. IfL 1 is diagonal thenτ ′ =τanda= 1γ.

Moreover, defining the norm

φ n :=|Φ| 2n := √ 1 n! |Φ| 2 ,forΦ∈ H n ,

we have the following lemma, proved in [7] (see Lemmas 2.10, 2.11):

Lemma 13 (i) Fork 1 ++k q =n |R q [Φ k 1 ,  ,Φ k q ]| 2 c ρ q φ 1   φ k q , n ≤ k

(ii) for2≤ℓ≤p,ℓ+k=n+ 1,and anyN k ∈ H k |DΦ ℓ N k | 2n ≤ℓ 2 + (m 0 −1)ℓ φ ℓ |N k | 2k ≤ℓ m 0 φ ℓ |N k | 2k  Then, the proof of Theorem 1 is performed in several steps giving re- spectively estimates ofφ n ,k P Φ(u 0 )k, andρ 0 gathered in the following lemmas:

10

Lemphin

LemSigmaphiopt

rhoopt

Lemma 14There existsK >0depending only onc, c 01 , ρ, m 0 , asuch that for everynwith1≤n≤p, φ n ≤m 0 K n−1 (n!) 1+τ ′ (16) wherec 01 :=max  |||P 0 |||,|||P 1 |||  . Lemma 15Let us choosepsuch that p=p opt :=  (2δ 1 K) b  , b= +1 1 τ ′ ,(17)

where[] fordenotes the integer part of a number. Then||u 0 || ≤δwe have X Φ k (u 0 )≤2δ √ m 0   1≤k≤p opt  Lemma 16The remainderρsatisfies

ρ(u 0 ) =R 1 (u 0 ) +R 2 (u 0 ) +R 3 (u 0 ) +R 4 (u 0 ),

with R 1 (u 0 ) = p+ X 1≤q P 1 R q      1≤ X k≤p Φ k (u 0 )   (q)    , R 2 (u 0 ) =− X DΦ ℓ [u 0 (ℓ) ]P 0 R q      1≤ X Φ k (u 0 )   (q)    , 2≤ℓ≤p p+1≤q k≤p R 3 (u 0 ) = X P 1 R q [Φ k 1 (u 0 ),  ,Φ k q (u 0 )], 2≤q≤p1≤k j ≤p k 1 ++k q ≥p+1 R 4 (u 0 ) =− X DΦ ℓ [u ( 0 ℓ) ]P 0 R q [Φ k 1 (u 0 ),  ,Φ k q (u 0 )], 2≤ℓ≤p q≤p1≤k j ≤p k 1 +k q ≥p−l+2 and forp=p opt , it satisfies sup||ρ(u 0 )|| ≤M e − δb ,(18) ||u 0 ||≤δ with >M, w0depending only onτ, m 0 , c, ρ,L 1 .

11

estimPhi na _

lowerBounda

Eqrhoa