Polynomial Normal Forms with Exponentially Small Remainder for Analytic Vector Fields

Gérard Iooss

Description

Polynomial Normal Forms with Exponentially Small Remainder for Analytic Vector Fields Gerard Iooss a,b and Eric Lombardi c,? aInstitut Non Lineaire de Nice, 1361 Routes des lucioles, O6560 Valbonne, France bInstitut Universitaire de France cInstitut Fourier, UMR5582, Laboratoire de mathematique. Universite de Grenoble 1, BP 74, 38402 Saint-Martin d'Heres cedex 2, France Abstract A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincare, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin, can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in Rm. More precisely we prove that for any vector field in Rm admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small.We also give several examples of non semi-simple linearization for which this result is still true.

• ?2 complex diagonal


• near identity


• analytic vector


• exponentially small


• vector fields near


• fields


• over exponentially


• jordan normal

Sujets

• Lombardi
• Fields
• Jordan normal form

Informations

• Publié par: pefav
• Nombre de lectures: 12
• Langue: English

Extrait

Polynomial Normal Forms with Exponentially Small Remainder for Analytic Vector Fields

G´erard Iooss ab and Eric Lombardi c ∗

a 1,63R1uoerediNeccioles,OtesdesluennoarF,0656blaVencInoNtutitsiae´niLn b Institut Universitaire de France c rGedboneelueiqni.Ursve´eiturieutFostitInarobaL,2855RMU,ratemh´atemedirto 1,BP74,38402Saint-Martind’H`erescedex2,France

Abstract

A key tool in the study of the dynamics of vector fields near an equilibrium point is thetheoryofnormalforms,inventedbyPoincar´e,whichgivessimpleformstowhich a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin, can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields inR m prove that for any vector field in. More precisely we R m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small.We also give several examples of non semi-simple linearization for which this result is still true.

Key words:Analytic vector fields, Normal forms, exponentially small remaiders

∗ Corresponding author. Fax: [phone] Email addresses:email [e-mail],)raIdooss(G´er [e-mail](Eric Lombardi). URLs:rn.snlc..wni//wwt/trpf:h∼iooss/Iords)os,G(are´ http://www-fourier.ujf-grenoble.fr/∼lombard/(Eric Lombardi).

Preprint submitted to Journal of differential equations

7 October 2004

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Introduction

1.1 Position of the problem

A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincare, which gives simple ´ forms to which a vector field can be reduced close to the equilibrium [1],[3]. In the class of formal vector valued vector fields the problem can be easily solved [1], whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs [3], [21],[22]. Neverthe- less the study of the dynamics in a neighborhood of the origin, can very often be carried out via a normalization up to finite order (see for instance [4], [11], [15], [16], [19],[23]). Normal forms are not unique and various characterization exist in the literature [2],[5],[8],[13],[23]. In this paper we will consider the version given in [13]:

Theorem 1.1 (Unperturbed NF-Theorem)LetVbe a smooth (resp. an- alytic) vector field defined on a neighborhood of the origin inR m (resp. in C m ) such thatV(0) = 0. Then, for any integerp≥2, there are polynomials Q p N p :R m →R m (resp.C m →C m ) , of degree≤p, satisfying Q p (0) =N p (0) = 0 DQ p (0) =DN p (0) = 0 such that under the near identity change of variableX=Y+Q p (Y), the vector field dX =V(X) (1)

dt becomes d d Y t =LY+N p (Y) +R p (Y) (2) whereDV(0) =L, where the remainderR p is a smooth (resp. analytic) func- tion satisfyingR p (X) =O(kXk p+1 )and where the normal form polynomial N p of degreepsatisfies

N p (e tL ∗ Y) = e tL ∗ N p (Y) for allY∈R m (resp. inC m ) andt∈Ror equivalently DN p (Y)L ∗ Y=L ∗ N p (Y) whereL ∗ is the adjoint ofL. Moreover, ifTis a unitary linear map which commutes withV, then for everyY,

Q p (T Y) =TQ p (Y)N p (T Y) =TN p (Y)

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Similarly, ifVrespect to some linear unitary symmetryis reversible with S (S=S −1 =S ∗ ), i.e. ifVanticommutes with this symmetry, then for every Y, Q p (SY) =SQ p (Y)N p (SY) =−SN p (Y)

This version of the Normal Form Theorem up to any finite order has the following advantages : its proof is elementary and the characterization given is global in terms of a unique commutation property. Moreover it uses a simple hermitian structure of the space of homogeneous polynomials of given degree.

Since a usual way to study the dynamics of vector fields close to an equilibrium is to see the full vector field as a perturbation of its normal formL+N p by higher order terms, it happens to be of great interest to obtain sharp upper bounds of the remaindersR p . A similar theory of resonant normal forms was developed for Hamiltonians systems written in action-angle coordinates (see for instance [6], [9], [20]). A sticking result obtained by Nekhoroshev [17], [18], in order to study the stability of the action variables over exponentially large interval of time, is that up to an optimal choice of the orderpof the normal form , the remainder can be made exponentially small. For more details of such Normal Form Theorems with exponentially small remainder for Hamiltonian systemswritteninactionanglevariables,werefertotheworkofP¨oschel in [20]. A similar result of exponential smallness of the remainder was also obtained by Giorgilli and Posilicano in [10] for areversible systemwith a linear part composed of harmonic oscillators.

So a natural question is to determine for which class of analytic vector fields, such results of normalization up to exponentially small remainder can be ob- tained?

Since in the previously mentioned works dealing with particular hamiltonian or reversible systems, the normalization procedure is based on diagonalizable homological operators, a first natural class to consider, is the classC s of an- alytic vector fields, fixing the origin, and such that their linearization at the origin issemi-simple(i.e. is diagonalizable). This is indeed the largest class for which the homological operators involved in the normalization procedure are diagonalizable (see Lemma 2.5-(a)). More precisely, we prove in this paper that such results of normalization up to exponentially small remainder can be obtained for any analytic vector fields inC s provided that the spectrum of the linearization at the origin satisfies some ”nonresonance assumptions” which enable to control the small divisor effects.

The question of the validity of such results for analytic vector fields with non semi-simple linearizationmore intricate : we give two examples ofis far non semi-simple linearizations for which the result is still true. However, the question remains totally open for other non semi simple linearizations. We

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perform some estimates which suggest that the results should not be true in general for non semi-simple linearizations, but theses estimates are not sufficient to build a counter example (see Remark 2.9).

1.2 Statement of the results

To state our results we need to specify some ”nonresonance assumptions” which enable to control the small divisor effects. In many problems, one uses one of the two following classical ”nonresonance assumptions” : for a subset ZofZ m , forK∈Nand forγ >0, a vectorλ= (λ 1    λ m )∈C m , is said to beγ K-nonresonant moduloZif for everyk∈Z m with|k| ≤K

| hλ ki | ≥γwhenk∈Z

(3)

Similarly, forγ >0 m > τ−1,λis said to beγ τ-Diophantine moduloZif for everyk∈Z m , | hλ ki |  |  ≥ γ k| τ whenk∈Z(4) where fork= (k 1    k m )∈Z m ,|k|:=|k 1 |+  +|k m |. However, in the problem of normal forms, the small divisors appear as eigenvalues of the ho- mological operator giving the normal forms by induction (see Subsection 2.1 and Lemma 2.5). To control these small divisors let us introduce two slightly different definitions :

Definition 1.2Let us defineλ= (λ 1   λ)∈C m ,K∈N > γ0and  m τ > m−1.

(a)The vectorλis said to beγ K-homologically without small divisors if for everyα∈N m with2≤ |α| ≤K, and everyj∈N,1≤j≤m,

| hλ αi −λ j | ≥γwhenhλ αi −λ j 6= 0 (b)The vectorλis said to beγ τ-homologically Diophantineif for every m α∈N,|α| ≥2, | hλ αi −λ j | ≥  | γ α| τ whenhλ αi −λ j 6= 0 (c)For a linear operatorLinR m , let us denote byλ 1    λ m its eigenval- ues andλ L := (λ 1    λ m ). ThenLis said to beγ K-homologically without small divisors( resp.γ τ-homologically Diophantine) if λ L is so. Remark 1.3Observe that in the above definitions, the components ofαare nonnegative integers whereas in (3) and (4),klies inZ m .

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In what follows we use Arnold’s notations [1] for denoting matrices under complex Jordan normal forms :λ 2 denotes the 2×2 complex Jordan block corresponding toλ∈Cwhereasλλrepresents 2×2 complex diagonal matrix diag(λ λ), i.e. λ 2 :=    0λ λ1    whereasλλ:=    λ0     0λ

A matrix under complex Jordan normal form is then denoted by the products of the name of its Jordan blocs. Moreover since for real matrices the Jordan blocks corresponding to non zero matrices occur by pairsλ r andλ r we shorten their name as follows : forλ 1  λ 2 ∈C\R,0 2 λ r 1 1 λ 2 r 2 λ r 1 1λ r 2 2is simply denoted by0 2 λ r 1 1 λ r 2 2 | C . Moreover, when one works with vector fields inR m , one may want to remain inR m and thus to use real Jordan normal forms for the linearization of the vector field. So, for∈Randλ=x+ iy∈C\R, we denote byµ 2 λ 2 | R the real Jordan matrix

 
0 1 ! 00 00 00 00 0 00 0
y x− x y !
1 0 0 1 !    00 00 00 00
y x− x y !  

Finally, we equipR m andC m with the canonical inner product and norm, i.e. m forX= (X 1    X m )∈C m ,kXk 2 :=hX Xi= P X j X j We are now j=1 ready to state our main result:

Theorem 1.4 (NF-Theorem with exponentially small remainder) LetVbe an analytic vector field in a neighborhood of0inR m (resp. inC m ) such thatV(0) = 0, i.e. V(X) =LX+ X V k [X (k) ] (5) k≥2 whereLis a linear operator inR m (resp. inC m ) and whereV k is bounded k-linear symmetric and

kV k [X 1    X k ]k ≤

with >c ρ0independent ofk.

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c kX 1 k  ρ  k  kX k k

(6)