Steklov–Lyapunov type systems ∗

A. Bolsinov and Yu. Fedorov

Department of Mathematics and Mechanics

Moscow Lomonosov University, Moscow, 119 899, Russia

bolsinov@mech.math.msu.su, fedorov@mech.math.msu.su

September 22, 2003

Abstract

In this paper we describe integrable generalizations of the classical Steklov–Lyapunov systems, which are defined on a certain product so(m) × so(m), aswell as the structure of rank r coadjoint orbits in so(m)×so(m). We show thatthe restriction of these systems onto some subvarieties of the orbits writtenin new matrix variables admits a new r × r matrix Lax representation in ageneralized Gaudin form with a rational spectral parameter.

In the case of rank 2 orbits a corresponding 2 × 2 Lax pair for the reducedsystems enables us to perform a separation of variables.

1 Introduction. Gaudin magnets and the hierarchy of

the Steklov–Lyapunov systems.

Many finite-dimensional integrable systems, as well as finite-gap reductions of someintegrable PDE’s, can be regarded as Hamiltonian flows on finite-dimensional coad-joint orbits of the loop algebra gl(r) described by r×r Lax equations with a spectralparameter λ ∈ C,

L(λ) = [ L(λ),M(λ) ] , L = Y +

n∑

i=1

Ni

λ − ai

, L,M ∈ gl(r), (1.1)

where Ni are r × r matrix variables, Y ∈ gl(r) is a constant matrix and a1, . . . , an

are arbitrary distinct constants (see [1, 2]). In particular, L(λ) can be taken in form

L(λ) = Y + GT (λIn − A)−1F (1.2)

where In is the n× n unit matrix and G, F are n× r matrices of rank r. Integrablesystems described by the corresponding Lax equations are usually referred to asGaudin magnets ( [8]) .

∗AMS Subject Classification 58F07, 70H99, 76B15

1

DOI 10.1007/s11005-006-0065-7

Letters in Mathematical Physics (2006) 76:111–134 © Springer 2006

Closed Geodesics and Billiards on Quadrics Related

to Elliptic KdV Solutions

SIMONETTA ABENDA1 and YURI FEDOROV2,3

1Dipartimento di Matematica e CIRAM, Universita degli Studi di Bologna, Bologna, Italy.

e-mail: abenda@ciram.unibo.it2Department of Mathematics and Mechanics, Moscow Lomonosov University,

Moscow 119 899, Russia.3Departament de Matematica I, Universitat Politecnica de Catalunya, Catalunya, Spain.

Received: 14 December 2004

Published online: 17 March 2006

Abstract. We consider algebraic geometrical properties of the integrable billiard on a quad-

ric Q with elastic impacts along another quadric confocal to Q. These properties are in

sharp contrast with those of the ellipsoidal Birkhoff billiards in Rn . Namely, generic com-

plex invariant manifolds are not Abelian varieties, and the billiard map is no more alge-

braic. A Poncelet-like theorem for such system is known. We give explicit sufficient condi-

tions both for closed geodesics and periodic billiard orbits on Q and discuss their relation

with the elliptic KdV solutions and elliptic Calogero system.

Mathematics Subject Classifications (2000). 37J60, 37J35, 70H45.

Keywords. geodesic flow, billiards, discrete integrable systems with deficiency, elliptic

Korteweg de Vries solutions, hyperelliptic tangential covers.

1. Introduction

One of the best known discrete integrable system is the billiard inside an n-dimen-

sional ellipsoid (more generally, a quadric)

Q =

{

X21

a1+· · ·+

X2n+1

an+1=1

}

⊆Rn+1, R

n= (X1, X2, . . . , Xn+1),

0<a1 < · · ·<an <an+1

with elastic reflections on Q [5]. The billiard inside Q (Birkhoff case) inherits the

remarkable property of geodesics on Q given by the Chasles theorem: the trajecto-

ries or their continuations before and after impacts are tangent to the same set of

n quadrics that are confocal to Q (see, e.g., [31]). The parameters of these quad-

rics deliver n independent and commuting first integrals of the discrete system.

Veselov [37] described the Birkhoff billiard map in terms of discrete Lagrang-

ian formalism and showed that its complex invariant manifolds are open subsets

of coverings of Jacobians of hyperelliptic curves. Moreover, the restriction of the

November 19, 2004 17:23 WSPC/Trim Size: 9in x 6in for Proceedings abenda˙qp

INTEGRABLE ELLIPSOIDAL BILLIARDS WITH

SEPARABLE POLYNOMIAL POTENTIALS

S. ABENDA

Dipartimento di Matematica e CIRAM

Universita degli Studi di Bologna, Italy

E-mail: abenda@ciram.unibo.it

YU. FEDOROV

Departament de Matematica Aplicada 3

Universitat Politecnica Catalunya, Spain

E-mail: yuri.fedorov@upc.es

In this paper we announce we found representation in Lax form and θ–functionalsolutions of an integrable real billiard system whose billiard map is given by a shifton a stratum of the Jacobi variety of a hyperelliptic curve. To our knowledge thisis the first example of integrable discrete system on a stratum solved explicitly.The proof of the main theorems will be published elsewhere.

1. Introduction

One of the best known classical discrete integrable systems, the ellipsoidal

billiard in Q0 =

{

x21

a1

+ · · · +x2

n

an

= 1

}

⊆ Rn with elastic reflections, can

be obtained as the limit of a geodesic flow on an n–dimensional ellipsoid,

Q ⊂ Rn+1 when one of the semiaxes tends to zero8 . Straight line trajecto-

ries of the ellipsoidal billiard inherit the remarkable property of geodesics

on the n–dimensional ellipsoid given by Chasles theorem: continuation of

trajectory before and after impacts are simultaneously tangent to n − 1

quadrics confocal to Q0.

On the other hand, there exists a hierarchy of integrable generalizations

of this problem (see, for instance Rauch-Wojciechowski and Tsiganov14 and

Kozlov and Treshev12): the particle moves on Q in the force field generated

by combinations of some basis polynomial potentials Vp of degree 2p, p ∈ N.

The continuous system on Q with potential Vp, as well as the corresponding

billiard limit, are completely integrable12.

687

Assessment of Mission Design Including Utilization of Libration

Points and Weak Stability Boundaries

Authors: E.Canalias, G.Gomez, M.Marcote, J.J.Masdemont. Academic Institution: Department de Matematica Aplicada, Universitat Politecnica de Catalunya and Department de Matematica Aplicada, Universitat de Barcellona. Approved by: Dario Izzo, Advanced Concepts Team (ESTEC)

Contacts:

Josep Masdemont e-mail: josep@barquins.upc.es Dario Izzo Tel: ++31 (0)71565 � 3511 Fax: ++31 (0)71565 � 8018 e-mail: act@esa.int

Ariadna id: 03/4103

Study length: 2 months. Contract Number: 18142/04/NL/MV

Available on the ACT net (www.esa.int/act)

The scattering map in the spatial Hill’s problem

A. Delshams∗, J.J. Masdemont†and P. Roldan‡

January 22, 2007

Abstract

We present a framework and methodology to compute the scattering

map associated to heteroclinic trajectories in the spatial Hill’s problem.

The scattering map can be applied to design of zero-cost transfer trajec-

tories in astrodynamics.

1 Introduction

The restricted three body problem describes the motion of an infinitessimal par-ticle (for instance, a spacecraft) under the gravitational influence of two massivebodies called primaries (for instance, the Sun and the Earth) that move in cir-cular orbits around their common center of mass [26]. The equations of motionhave five equilibrium points, called libration points and denoted L1, L2, . . . L5.Hill’s problem is a simplified version of the classical restricted three body prob-lem that is useful for studying motion in the neighborhood of the equilibriumpoints L1 and L2. The model is only valid when the mass of one primary issmall compared with that of the other primary. Hill’s problem is described insection 2. In contrast with other papers, we use the spatial Hill’s model, thatis, the motion of the particle is not restricted to the plane of motion of theprimaries.

We study transfers of the infinitessimal particle from the neighborhood of L1

to that of L2 or viceversa. First we consider the normally hyperbolic invariantmanifolds associated to the equilibrium points. See section 3.3 for the definitionof a normally hyperbolic invariant manifold (NHIM for short). Then we computethe asymptotic behavior of heteroclinic solutions from one NHIM to the other.

The scattering map is a map between the normally hyperbolic invariantmanifolds that describes the asymptotic behavior of heteroclinic solutions. It isexplained in section 5.3. For instance, the scattering map answers the followingquestions: Does a heteroclinic solution tend to a periodic orbit? To which orbitdoes it tend to? What specific solution does it synchronize with?

The goal of this paper is to compute the scattering map associated to hete-roclinic trajectories from the neighborhood of L1 to the neighborhood of L2 (orvice versa) in the spatial Hill’s problem.

As an example application, consider a spacecraft orbiting in a given Lissajousorbit with small out-of-plane amplitude around L1 in the Sun-Earth system. For

∗<Amadeu.Delshams@upc.edu>

†<josep@barquins.upc.edu>

‡<Pablo.Roldan@upc.edu>, corresponding author

1

November 19, 2004 17:30 WSPC/Trim Size: 9in x 6in for Proceedings fontich

NON-RESONANT INVARIANT MANIFOLDS IN

NON-UNIFORMLY HYPERBOLIC SYSTEMS

ERNEST FONTICH

Departament de Matematica Aplicada i Analisi

Universitat de Barcelona

Gran Via, 585, 08007 Barcelona, Spain

RAFAEL DE LA LLAVE

Department of Mathematics

The University of Texas at Austin

Austin, TX 78712-1082, U.S.A

PAU MARTIN

Departament de Matematica Aplicada IV

Universitat Politecnica de Catalunya

Ed-C3, Jordi Girona, 1-3, 08034 Barcelona Spain

Let {xi}i∈N be a regular orbit of a C2 dynamical system f . Let S be a subset of itsLyapunov exponents. Assume that all the Lyapunov exponents in S are negativeand that the sums of Lyapunov exponents in S do not agree with any Lyapunovexponent in the complement of S. Denote by ES

xithe linear spaces spanned by

the spaces associated to the Lyapunov exponents in S. We show that there aresmooth manifolds W S

xisuch that f(W S

xi) ⊂ W S

xi+1and Txi

W Sxi

= ESxi

.

1. Introduction

In the study of the behavior of a dynamical system f around an orbit,

the first step is usually the study of the linearized system. The tangent

space at points of the orbit can be splitted into invariant subspaces corre-

sponding to different rates of exponential growth, which are related to its

Lyapunov exponents. Then, a natural question is the existence of nonlinear

counterparts to these invariant subspaces.

Here we shall consider orbits whose tangent spaces admit a well defined

splitting associated to different rates of exponential growth. Oseledec’s

theorem ensures that these orbits form a set of large measure. When the

877

Ergod. Th. & Dynam. Sys. (First published online 2005), 0, 1–20∗

doi:10.1017/S0143385704001038 c© 2005 Cambridge University Press∗Provisional—final page numbers to be inserted when paper edition is published

Algebraic proof of the non-integrability

of Hill’s problem

JUAN J. MORALES-RUIZ†, CARLES SIMO‡ and SERGI SIMON‡

† Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya,

Pau Gargallo, 5, 08028 Barcelona, Spain

(e-mail: juan.morales-ruiz@upc.es)

‡ Departament de Matematica Aplicada i Analisi, Universitat de Barcelona,

Gran Via, 585, 08007 Barcelona, Spain

(e-mail: carles@maia.ub.es, sergi@mat.ub.es)

(Received 22 September 2004 and accepted in revised form 11 November 2004)

Abstract. Hill’s lunar problem appears in celestial mechanics as a limit of the restricted

three-body problem. It is parameter-free and thus globally far from any simple well-known

problem, and has shed strong numerical evidence of its lack of integrability in the past.

An algebraic proof of meromorphic non-integrability is presented here. Beyond the result

itself, the paper can also be considered as an example of the application of differential

Galois and Morales–Ramis theories to a significant problem.

1. Introduction

Hill’s problem (HP), usually dubbed lunar as a homage to its earliest motivation, or planar

in order to distinguish it from its own extension to R3, is a model originally based on the

Moon’s motion under the joint influence of Earth and Sun [4]. A first simplification of

the general three-body problem consists in assuming the Moon’s mass is negligible and

the primaries (Earth and Sun) move in circular orbits around their common barycentre.

We then have a Hamiltonian system called the restricted three-body problem (RTBP,

see [16]). Let µ = ME/(ME + MS) and use a rotating coordinate frame whose first

axis is spanned by the primaries. By a suitable choice of mass, length and time units we

obtain the best-known equations for the RTBP:

ξ ′′ − 2η′ = �ξ ,

η′′ + 2ξ ′ = �η,

}

�(ξ, η) =1 − µ

r1+

µ

r2+

1

2(ξ2 + η2),

{

r21 = (ξ − µ)2 + η2,

r22 = (ξ − µ + 1)2 + η2,

that is, � is the gravitational plus the centrifugal potential. Setting the Earth as the origin

of coordinates and scaling length by µ1/3, HP is now defined by taking µ → 0 in the

resulting equations. Thus, the RTBP can be written as an O(µ1/3) perturbation of HP

Integrability of Hamiltonian Systems and Differential Galois

Groups of Higher Variational Equations

Juan J. Morales-RuizDepartament de Matematica Aplicada II

Universitat Politecnica de CatalunyaEdifici Omega, Campus Nord

c/ Jordi Girona, 1-3, E-08034 Barcelona, SpainE-mail: Juan.Morales-Ruiz@upc.es

Jean-Pierre RamisInstitut Universitaire de France and Laboratoire Emile Picard

Universite Paul Sabatier118, route de Narbonne, Toulouse, Cedex, France

E-mail: ramis@picard.ups-tlse.fr

Carles SimoDepartament de Matematica Aplicada i Analisi

Universitat de BarcelonaGran Via 585, E-08007 Barcelona, Spain

E-mail: carles@maia.ub.es

June 16, 2005

Abstract

Given a complex analytical Hamiltonian system, we prove that a necessary con-dition for meromorphic complete integrability is that the identity component of theGalois group of each variational equation of arbitrary order along each integral curvemust be commutative. This was conjectured by the first author based on a suggestionmade by the third author due to numerical and analytical evidences concerning higherorder variational equations. This non-integrability criterion extends to higher orders anon-integrability criterion (Morales-Ramis criterion), using only the first order varia-tional equation, obtained by the first and the second author. Using our result (at ordertwo, three or higher) it is possible to solve important open problems of integrabilitywhich escaped to Morales-Ramis criterion.

1 Introduction

The problem of integrability by quadratures, or in closed form, of dynamical systems is avery old, important and difficult problem. We know that, given an algebraic or analytic

dynamical system, defined by ordinary differential equations, a solution always existslocally and sometimes we can prolong it for every time towards the past and the future.Then, as the mathematicians of XVIII-th century, we would like to find the general solutionanalytically in an “explicit” way. When this is possible we can say that the system is

1

Estabilitat efectiva i tors invariants

de sistemes hamiltonians quasi-integrables

Pere Gutierrez i Serres

Memoria presentada per aspirar al

grau de Doctor en Ciencies Matematiques

Departament de Matematica Aplicada i Analisi

Universitat de Barcelona

Barcelona, abril de 1995

LA GACETA DE LA RSME, Vol. 6.2 (2003), Pags. 305–348 305

La publicacion electronica de la investigacion matematica:Apuntes para un debate sobre su economıa y sociologıa

por

Rafael de la Llave

Proponemos algunas reflexiones sobre la economıa de los medios electroni-

cos de comunicacion y su relevancia para la publicacion de la investiga-

cion matematica. Tratamos de apuntar algunos problemas economicos y

sociologicos para el futuro inmediato.

1. INTRODUCCION

Hace unos anos escribı un ensayo acerca de la publicacion electronica enmatematicas poniendo el enfasis sobre todo en cuestiones tecnicas1 .

En aquel ensayo concluımos que los problemas tecnicos, para que la dis-tribucion electronica se convirtiera en la manera dominante de distribucion dela investicacion matematica, eran facilmente superables. No ası los problemaseconomicos y sociologicos. Mas aun, la forma en la que la tecnologıa se iba aaplicar no estaba nada clara.

Esto es precisamente lo que quiero estudiar en estas notas: ¿Cuales son losproblemas economicos y de organizacion que la nueva tecnologıa pone sobrela mesa para la publicacion de la investigacion matematica23.

Creo que hay que recalcar que esta reflexion es necesaria. Cuando se in-troduce una nueva tecnologıa, casi siempre da lugar a consecuencias que soninesperadas. Un ejercicio muy interesante es leer lo que la gente decıa de lapıldora, de la television, del telefono o del coche cuando aparecieron. Serıapresuntuoso pensar que vamos a estar ahora mas acertados cuando hablamosde internet o del movil.

Considero que la unica posibilidad de que las cosas acaben bien es quehaya una discusion muy amplia e informada y que amplios grupos de personas

1Se puede encontrar una copia de dicho artıculo en

ftp.ma.utexas.edu://pub/papers/llave/publicacion.ps.

Contiene bastantes referencias a diversas fuentes que son relevantes en este artıculo; es por

ello que apenas citare documentos para respaldar algunas conclusiones tecnicas. Aunque

dicho ensayo no esta al dıa y hay muchas tecnologıas allı discutidas que ya son obsoletas,

creo que las conclusiones de la viabilidad se han reforzado.2Hay otros problemas relacionados que no abordaremos en este artıculo; por ejemplo, la

publicacion de libros de texto o otros medios de comunicacion de investigacion, e.g. telecon-

ferencia, correo electronico que no son publicacion.3Cuando se mencione publicacion matematica en el texto siguiente debe entenderse el

mas farragoso publicacion de la investigacion en matematicas.

On reductions of geodesic flows with left- or

right-invariant metrics onto Lie groups and their

symmetry fields

M. V. Deryabin, Yu. N. Fedorov

Department of Mathematics and Mechanics

Moscow Lomonosov University, Moscow, 119 899, Russia

M.Deriabine@mat.dtu.dk, fedorov@mech.math.msu.su

February 23, 2007

Abstract

In this paper we consider geodesic flows of left- or right-invariant metrics onarbitrary Lie groups and the corresponding Euler equations on their coalgebras.For a fixed value of momentum we describe the reduced vector field on thegroup, as well as its (right or left-invariant) symmetry fields. As examples, westudy an analog of the Euler top on the group SO(n) and the Euler equationsof ideal hydrodynamics on an infinite-dimensional group SDiff(M) of volumepreserving diffeomorphisms of an n-dimensional manifold M .

1 Introduction

For the classical Euler top there is the following remarkable reduction procedure: byfixing constants of Noether integrals, the vector field on the coalgebra so

∗(3) can beuniquely projected onto the group SO(3) itself (see, for example, [14]). The reducedflow on the group SO(3) was studied in [9], [10] (see also [11] for some generalizationsof these results to a finite-dimensional Lie group with left- or right-invariant metric).

We prove a general theorem on reduction of a geodesic flow of a left- or right-invariant metric onto a Lie group. For any Lie group (including those that are notBanach manifolds, like a group of volume preserving diffeomorphisms SDiff(M) ofa manifold M) we find both the reduced vector field and its ”symmetry fields”, i.e.,left- or right-invariant fields on the group that commute with our reduced vectorfield. As examples, we consider the n-dimensional Euler top and Euler equations ofideal hydrodynamics.

This approach has many advantages in comparison with traditional methods thatrequire solving the kinematic Euler equations in order to determine the trajectoryon the group. Besides it allows one to bypass the studying of coadjoint orbits, whichin many cases appears to be very complicated (for example, in problems of idealhydrodynamics, see [1]).

1

Review on the book

Galois Theory of Linear Differential Equations by Marius van der Put andMichael Singer, Springer, Berlin 2003, ISBN 3-540-44228-6.

The Differential Galois Theory is the Galois theory for differential equations.This theory has their origins in the works of Picard ([21, 22], [23] Chapitre XVII)and Vessiot ([28]) at the end of the XIX century about the Galois theory of lineardifferential equations, for this reason the Galois theory of linear differentialequations is also called the Picard-Vessiot theory.

The Picard-Vessiot theory was translated in the modern language of ex-tensions of differential fields by Kolchin in the middle of the XX century (see[14, 11], and references therein). In a similar way to the classical Galois theoryof polynomials, one start with a differential field K of coefficients of the lineardifferential equation

y′ = Ay, (1)

A ∈ Mat(n,K). Then we consider the smallest differential field L that containsK and all the solutions of the linear differential equation above. Kolchin provedthe existence and unicity of the extension L/K, this extension was named byKolchin the Picard-Vessiot extension associated to the linear differential equa-tion, provided that the characteristic of K is zero and has an algebraic closedfield of constants. Like in the Galois theory of polynomials, the Galois groupof equation (1), G = Gal(L/K), is defined as the set of (differential) automor-phisms of L which leaves fixed the coefficient field K; furthermore G is a linearalgebraic group. Another important result is that the Galois group gives a char-acterization of the linear equations that can be integrated in “closed form”: theidentity component of the Galois group must be a solvable group; I call this kindof equations integrable. Closed form solutions means that the general solution isobtained by a combination of algebraic functions, quadratures and exponentialof quadratures; this is similar to the Galois theorem about the solvability of apolynomial equation by radicals. Kolchin also extends the differential Galoistheory to some special non-linear differential equations, in such a way that theassociated differential extensions have nice normality properties; these exten-sions are named by Kolchin the strongly normal extensions ([14]) . From thecomplex algebraic-geometric point of view, the structure of the strongly normalextensions was studied by Buium ([8]). Roughly spiking strongly normal exten-sions of the complex field are given by solutions of linear differential equationswith coefficients in a field of abelian functions.

From the forties to the seventies of the XX century, the differential Galoistheory was essentially only studied by Kolchin’s school and it seems that duringseveral years the small but nice book [11] was the only available monograph onthe subject and contributed in a essential way to its diffusion. In the late 1970’s

1

Physica D 217 (2006) 107–120

www.elsevier.com/locate/physd

On the numerical computation of Diophantine rotation numbers of analyticcircle maps

Tere M. Seara∗, Jordi Villanueva

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Received 15 September 2005; received in revised form 2 March 2006; accepted 13 March 2006

Available online 5 May 2006

Communicated by J. Stark

Abstract

In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus

on analytic circle diffeomorphisms, but the method also works in the case of (enough) finite differentiability. The keystone of the method is that,

under these conditions, the map is conjugate to a rigid rotation of the circle. Moreover, although it is not fully justified by our construction, the

method turns out to be quite efficient for computing rational rotation numbers. We discuss the method through several numerical examples.

c© 2006 Elsevier B.V. All rights reserved.

Keywords: Circle maps; Rotation number; Numerical approximation

1. Introduction

The main purpose of this work is to introduce a new

numerical method to compute the rotation number of a circle

map. This problem has been formerly considered by many other

authors, and several algorithms have been developed. See, for

instance, [32,5,21,25,24,4,12,13,8]. On the one hand, the level

of complexity of these algorithms ranges from the definition

itself to sophisticated methods of frequency analysis. On the

other hand, some of them are efficient for the computation

of rational rotation numbers and some others work better for

irrational ones.In this paper we are mainly concerned with analytic circle

diffeomorphisms having Diophantine rotation number. So, we

take strong advantage of the fact that the map is analytically

conjugate to a rotation. The method we present is based on

the computation of suitable averages of the iterates of the map,

followed by Richardson’s extrapolation. The keystone of this

procedure is that we know a priori which is the asymptotic

behavior of these averages when the number of iterates goes

to infinity. This algorithm provides numerical approximations

to the rotation number, with very high accuracy in general.

∗ Corresponding author. Tel.: +34 934016553; fax: +34 934011713.E-mail addresses: tere.m-seara@upc.edu (T.M. Seara),

jordi.villanueva@upc.edu (J. Villanueva).

To develop this method, we use the hypotheses on the map

to be analytically conjugate to a rigid rotation and to have a

(good) Diophantine rotation number. Although we focus on the

analytic case, the same procedure can be used for smooth circle

diffeomorphisms, but we only expect the method to be efficient

if the conjugation is regular enough.

Of course, the set up of this method is restrictive and

excludes a lot of cases. For instance, if we consider a (generic)

one-parameter family of circle homeomorphisms, the set of

parameters for which the rotation number is rational, and hence

the map is not conjugate to a rotation (in general), is a dense set

with (non-empty) interior. However, if these maps are smooth

perturbations of a rotation, then, under general hypotheses, the

set of parameters for which the rotation number is Diophantine

has big relative measure. On the other hand, if the rotation

number is eventually rational, the method provides quite good

results. We do not have a complete justification of this fact, but

we refer to Remark 9 for a tentative explanation and to Section 4

for examples with rational rotation numbers.

From the practical point of view, the numerical method

presented here is suitable if we are able to compute the iterates

of the map with high precision, for instance if we can work

with a computer arithmetic having a large number of decimal

digits. In this case, we can try to use the method with high-

order extrapolation and, then, we can hope to obtain a good

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.03.013

Butlletí de la Societat Catalana de Matemàtiques

Vol. 18, núm. 1, 2003. Pàg. 129–151

Resumació de Borel i teoria de la ressurgència

Tere M. Seara i David Sauzin

1 Introducció

Aquestes notes estan basades en unes xerrades que van tenir lloc al seminari

de sistemes dinàmics de la Universitat Autònoma de Barcelona l’any 2002. El

que volíem en aquelles xerrades i també en aquest article és relacionar els

mètodes clàssics de resumació de Borel de sèries divergents (que introduirem

seguint la presentació de [3]) i la més moderna teoria de la ressurgència i

veure com ambdós són eines útils en alguns problemes de gran rellevància en

la moderna teoria de sistemes dinàmics.

La idea que les sèries divergents poden ser iguals d’útils o més que les sè-

ries amb radi de convergència positiu ja ve de molt antic, com podem veure

a la citació següent de Poincaré: «hi ha entre els geòmetres i els astrònoms

una mena de malentès respecte el significat de la paraula convergència. Els

geòmetres, preocupats pel perfecte rigor i sovint molt indiferents a la llarga-

da dels càlculs dels quals en veuen la possibilitat sense fer-los efectivament,

diuen que una sèrie és convergent quan la suma dels termes tendeix a un lí-

mit determinat, encara que els primers termes disminueixin molt lentament.

Els astrònoms, al contrari, tenen costum de dir que una sèrie convergeix quan

els vint primers termes, per exemple, disminueixen molt ràpidament, encara

que els termes següents podrien créixer indefinidament. Així, per prendre un

exemple simple, considerem les dues sèries que tenen per terme general

1000n

n!i

n!

1000n.

»Els geòmetres dirien que la primera sèrie convergeix, i de fet que convergeix

ràpidament, perquè el terme mil.lionèssim és molt més petit que el 999999e;

però veurien la segona com divergent, perquè el terme general creix indefini-

dament.

CALCUL DE L’ESCISSIO DE

SEPARATRIUS USANT TECNIQUES

DE MATCHING COMPLEX I

RESSURGENCIA APLICADES A

L’EQUACIO DE

HAMILTON-JACOBI

Carme Olive Farre

Departament d’Enginyeria Informatica i Matematiques

Universitat Rovira i Virgili

17 de maig de 2006

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