Hamiltonian Systems withThree or More Degrees of Freedom

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Series C: Mathematical and Physical Sciences - Vol. 533

Hamiltonian Systems with Three or More Degrees of Freedom edited by

Carles Sim6 Universitat de Barcelona, Barcelona, Spain

.... " Springer-Science+Business Media, B.V.

Proceedings of the NATO Advanced Study Institute on Hamiltonian Systems with Three or More Oegrees of Freedom S'Agar6, Spain June 19-30, 1995

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-5968-8 ISBN 978-94-011-4873-9 (eBook) DOI 10.1007/978-94-011-4673-9

Printed on acid-free paper

AII Rights Reserved 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1 st edition 1999 No part of the material protected by this copyright notice may be reproduced ar utilized in any form or by any means, electronic or mechanical, including phatocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Table of contents

LECTURES

S. Angenent

INFLECTION POINTS, EXTATIC POINTS AND CURVESHORTENING

V.I. Arnold

TOPOLOGICALLY NECESSARY SINGULARITIES ON MOVINGWAVEFRONTS AND CAUSTICS

S.V. Bolotin

HETEROCLINIC CHAINS OF SKEW PRODUCT HAMILTONIANSYSTEMS

G. Contopoulos, N. Voglis, C. Efthymiopoulos

ORDER AND CHAOS IN 3-D SYSTEMS

A. Delshams, R. Ramirez-Ros, T.M. Seara

SPLITTING OF SEPARATRICES IN HAMILTONIAN SYSTEMSAND SYMPLECTIC MAPS

L.R. Eliasson

ON THE DISCRETE ONE-DIMENSIONAL QUASI-PERIODICSCHRODINGER EQUATION AND OTHER SMOOTHQUASI-PERIODIC SKEW PRODUCTS

G. Gallavotti

LINDSTEDT SERIES AND KOLMOGOROV THEOREM

A. Giorgilli, U. Locatelli

A CLASSICAL SELF-CONTAINED PROOF OF KOLMOGOROV'STHEOREM ON INVARIANT TORI

3

11

13

26

39

55

62

72

vi

P. Boyland, C. Gole

DYNAMICAL STABILITY IN LAGRANGIAN SYSTEMS

90

J. Henrard 115

THE ORIGIN OF CHAOTIC BEHAVIOUR IN THE KIRKWOODGAPS

M. Herman

EXAMPLES OF COMPACT HYPERSURFACES IN R2P , 2P ~ 6,WITH NO PERIODIC ORBITS

V.V. Kozlov

HAMILTONIAN SYSTEMS WITH THREE DEGREES OFFREEDOM AND HYDRODYNAMICS

J. Laskar

INTRODUCTION TO FREQUENCY MAP ANALYSIS

A. Jorba, R. de la Llave, M. ZouLINDSTEDT SERIES FOR LOWER DIMENSIONAL TORI

P. Lochak

ARNOLD DIFFUSION; A COMPENDIUM OF REMARKSAND QUESTIONS

J. Moser

OLD AND NEW APPLICATIONS OF KAM THEORY

126

127

134

151

168

184

A. Neishtadt 193

ON ADIABATIC INVARIANCE IN TWO-FREQUENCY SYSTEMS

J. Seimenis

THE METHOD OF RATIONAL APPROXIMATIONS:THEORY AND APPLICATIONS

C. Sim6

DYNAMICAL SYSTEMS METHODS FOR SPACE MISSIONSON A VICINITY OF COLLINEAR LIBRATION POINTS

213

223

Va. G. Sinai

A MECHANISM OF ERGODICITY IN STANDARD--LIKE MAPS

D.V. Treschev

CONTINUOUS AVERAGING IN HAMILTONIAN SYSTEMS

S. Wiggins

PHASE SPACE GEOMETRY AND DYNAMICS ASSOCIATEDWITH THE 1 : 2 : 2 RESONANCE

H. Yoshida

FROM SINGULAR POINT ANALYSIS TO RIGOROUS RESULTSON INTEGRABILITY: A DREAM OF S. KOWALEVSKAYA

CONTRIBUTIONS

S. Abenda

TIME SINGULARITIES FOR POLYNOMIAL HAMILTONIANSWITH ANALYTIC TIME DEPENDENCE

M. Antoni, C. Sandoz, Y. Elskens

vii

242

244

254

270

285

290

NUMERICAL STUDY OF TURBULENCE IN N-BODY HAMILTON-IAN SYSTEMS WITH LONG RANGE FORCE I. RELAXATION

K.M. Atkins, M. Hutson

PHASE SPACE STRUCTURES IN 3 AND 4 DEGREES OFFREEDOM: APPLICATION TO CHEMICAL REACTIONS

A. Bazzani, F. Brini

MODULATED DIFFUSION FOR SYMPLECTIC MAPS

G. Benettin

ON THE JEANS-LANDAU-TELLER APPROXIMATION FORADIABATIC INVARIANTS

295

300

305

viii

R.M. Benito, F. Borondo

PERIODIC ORBITS AND QUANTUM MECHANICS OFMOLECULAR HAMILTONIAN SYSTEMS

310

F. Borondo, J. Bowers, R. Guantes, Ch. Jaffe, S. Miret-Artes 314

CHAOS IN ATOM-SURFACE COLLISIONS

T. Bountis, L. Drossos

ON THE NON-INTEGRABILITY OF THE MIXMASTERUNIVERSE MODEL

S.A. Dovbysh

BRANCHING OF SOLUTIONS AS OBSTRUCTION TO THEEXISTENCE OF A MEROMORPHIC INTEGRAL IN MANY-DIMENSIONAL SYSTEMS

H.R. Dullin

THE ENERGY SURFACES OF THE KOVALEVSKAYA-TOP

H.S. Dumas

FILLING RATES FOR LINEAR FLOW ON THE TORUS:RECENT PROGRESS AND APPLICATIONS

N. Voglis, C. Efthymiopoulos, G. Contopoulos

INVARIANT SPECTRA OF ORBITS IN MULTIDIMENSIONALSYMPLECTIC MAPS

L. Chierchia, C. Falcolini

ON THE CONVERGENCE OF FORMAL SERIES CONTAININGSMALL DIVISORS

Y.N. Fedorov

SYSTEMS WITH AN INVARIANT MEASURE ON LIE GROUPS

S. Ferraz-Mello

STOCHASTICITY OF THE 2/1 ASTEROIDAL RESONANCE:A SYMPLECTIC MAPPING APPROACH

318

324

330

335

340

345

350

357

S. Ferrer, M. Lara, J. Palacian, P. Yanguas

ix

362

ON PERTURBED OSCILLATORS IN 1-1 - 1 RESONANCE: CRIT-ICAL INCLINATION IN THE 3D HENON-HEILES POTENTIAL

A. Delshams, Y. Gelfreich, A. Jorba, M.T. Seara

SPLITTING OF SEPARATRICES FOR (FAST) QUASIPERIODICFORCING

G. Gentile, Y. Mastropietro

A POSSIBLE MECHANISM FOR THE KAM TORI BREAKDOWN

c. Grotta RagazzoON THE DYNAMICS NEAR RESONANT EQUILIBRIA

A. Delshams, P. Gutierrez

EXPONENTIALLY SMALL ESTIMATES FOR KAM THEOREMNEAR AN ELLIPTIC EQUILIBRIUM POINT

G. Haller

FAST DIFFUSION AND UNIVERSALITY NEAR INTERSECTINGRESONANCES

H. Han13mann

QUASI-PERIODIC MOTION OF A RIGID BODY UNDER WEAKFORCES

A.Haro

367

372

377

386

391

398

403

CENTER AND CENTER-(UN)STABLE MANIFOLDS OF ELLIPTIC-HYPERBOLIC FIXED POINTS OF 4D-SYMPLECTIC MAPS. ANEXAMPLE: THE FROESCHLE MAP

P.G. Hjorth

A MANY-PARTICLE ADIABATIC INVARIANT

S. Ichtiaroglou, E. Meletlidou

THE NON-INTEGRABILITY OF PERTURBED HAMILTONIANSOF N DEGREES OF FREEDOM AND THE CONTINUATION OFPERIODIC ORBITS

408

413

x

M.Irigoyen

STRUCTURE OF THE 5-DIMENSIONAL MANIFOLD OFCONSTANT ENERGY IN THE h-PROBLEM

418

J.A. Steckel, Ch. Jaffe 422

THE BIFURCATIONS OF THE LANGMUIR ORBIT IN THETWO-ELECTRON ATOM

H.R. Jauslin, M. Govin, M. Cibils 426

DOMAINS OF CONVERGENCE OF KAM TYPE ITERATIONSFOR EIGENVALUE PROBLEMS

A.. Jorba, J. Masdemont 430NONLINEAR DYNAMICS IN AN EXTENDED NEIGHBOURHOODOF THE TRANSLUNAR EQUILIBRIUM POINT

S. Keshavamurthy, G.S. Ezra 435

ANALYSIS OF QUANTUM EIGENSTATES IN A 3-MODE SYSTEM

T. Konishi

DETECTION OF STABLE MANIFOLDS IN HIGH DIMENSIONALPHASE SPACE

T. Konishi

440

444

ORDERED MOTION IN HAMILTONIAN SYSTEMS WITH MANYDEGREES OF FREEDOM

I.Yu. Kostyukov, G.M. F'raiman

SOME RELATIONS FROM HAMILTONIAN MECHANICS ANDTHEIR APPLICATIONS TO PLASMA PHYSICS

A. Delshams, J.T. Lazaro

EFFECTIVE STABILITY IN REVERSIBLE SYSTEMS

O.Yu. Koltsova, L.M. Lerman

NEW CRITERION OF NONINTEGRABILITY FOR ANN-DEGREES-OF-FREEDOM HAMILTONIAN SYSTEM

449

453

458

M.W.Lo

THE INVARIANT MEASURE FOR THE SATELLITE GROUNDSTATION VIEW PERIOD PROBLEM

A.J. Maciejewski

THE TWO RIGID BODIES PROBLEM. REDUCTION ANDRELATIVE EQUILIBRIA

J.P. Marco

TRANSITION ORBITS AND TRANSITION TIMES ALONGCHAINS OF HYPERBOLIC TORI

J. Martinez, R. Lopez

PERSISTENCE OF ASTEROIDS AFTER A CLOSE ENCOUNTER

M. Matveyev

STRUCTURE OF THE SETS OF INVARIANT TORI ANDPROBLEMS OF STABILITY IN REVERSIBLE SYSTEMS

J.D. Meiss

ON THE BREAK-UP OF INVARIANT TORI WITH THREEFREQUENCIES

J. von Milczewski, G.H.F. Diercksen, T. Uzer

THE ARNOL'D WEB IN ATOMIC PHYSICS

A. Mir, A. Delshams

PSI-SERIES, SINGULARITIES OF SOLUTIONS ANDINTEGRABILITY OF POLYNOMIAL SYSTEMS

J.J. Morales-Ruiz, J.P. Ramis

GALOISIAN OBSTRUCTIONS TO INTEGRABILITY OFHAMILTONIAN SYSTEMS: STATEMENTS AND EXAMPLES

A. Morbidelli

BOUNDS ON DIFFUSION IN PHASE SPACE: CONNECTIONBETWEEN NEKHOROSHEV AND KAM THEOREMS ANDSUPEREXPONENTIAL STABILITY OF INVARIANT TORI

Xl

471

475

480

485

489

494

499

504

509

514

xii

M. OUe, D. Pfenniger

BIFURCATION AT COMPLEX INSTABILITY

Y. Papaphilippou

GLOBAL DYNAMICS OF A GALACTIC POTENTIAL VIAFREQUENCY MAP ANALYSIS

R. Ramirez-Inostroza, R. Masip

A PARTICULAR CLASS OF INTEGRABLE CONNECTION

A. Delshams, R. Ramirez-Ros

POINCARE-MELNIKOV-ARNOLD METHOD FOR TWIST MAPS

V. Rom-Kedar

TRANSPORT IN A CLASS OF N-D.O.F. SYSTEMS

V.M. Rothos, T.C. Bountis

518

523

528

533

538

544

MEL'NIKOV VECTOR AND SINGULARITY ANALYSIS OFPERIODICALLY PERTURBED 2 D.O.F HAMILTONIAN SYSTEMS

L. Sbano

A STUDY OF REDUCED ACTION-FUNCTIONAL FOR THENEWTONIAN 3-BODY PROBLEM

D.J. Scheeres

SATELLITE DYNAMICS ABOUT ASTEROIDS: COMPUTINGPOINCARE MAPS FOR THE GENERAL CASE

A. Calini, C.M. Schober

549

554

558

MEL'NIKOV ANALYSIS OF A HAMILTONIAN PERTURBATIONOF THE NONLINEAR SCHRODINGER EQUATION

C.G. Schroer

FIRST ORDER ADIABATIC APPROXIMATION FOR A CLASSOF CLASSICAL SLOW-FAST SYSTEMS WITH ERGODIC FASTDYNAMICS

563

xiii

M.B. Sevryuk 568

THE LACK-OF-PARAMETERS PROBLEM IN THE KAM THEORYREVISITED

D.K. Sharomov 573

HOMOCLINIC INVARIANT FOR THE COUPLED STANDARD MAP

V.V. Sidorenko 578

DESTRUCTION OF ADIABATIC INVARIANTS ON RESONANCES:EXAMPLE FROM THE RIGID BODY DYNAMICS

Ch. Skokos, G. Contopoulos, C. Polymilis

NUMERICAL STUDY OF THE PHASE SPACE OF A FOURDIMENSIONAL SYMPLECTIC MAP

T.J. Stuchi, R. Vieira-Martins, R. Rodrigues

CAUSTICS, CUSPS AND CHAOS IN HAMILTONIAN SYSTEMS

M.B. Tabanov

A HOMOCLINIC ORBIT FOR THE DOUBLE PENDULUM

J.C. Tatjer

SOME BIFURCATIONS RELATED TO HOMOCLINICTANGENCIES FOR I-PARAMETER FAMILIES OFSYMPLECTIC DIFFEOMORPHISMS

E. Todesco, M. Gemmi

RESONANT NORMAL FORMS FOR FOUR DIMENSIONALSYMPLECTIC MAPPINGS AND APPLICATIONS TONONLINEAR BEAM DYNAMICS

S. Tompaidis

APPROXIMATION OF INVARIANT SURFACES BY PERIODICORBITS IN HIGH-DIMENSIONAL MAPS

A. Tovbis, M. Tsuchiya, Ch. Jaffe

EXPONENTIAL ASYMPTOTICS AND APPROXIMATION OFSTABLE AND UNSTABLE MANIFOLDS IN SINGULARLYPERTURBED NONLINEAR SYSTEMS

583

588

592

595

600

605

610

xiv

K. Umeno

SINGULARITY ANALYSIS TOWARDS NONINTEGRABILITYOF NONHOMOGENEOUS NONLINEAR LATTICES

A.I. Neishtadt, D.L. Vainshtein, A.A. Vasiliev

ADIABATIC CHAOS OF STREAMLINES IN A FAMILY OF 3DCONFINED STOKES FLOWS

614

618

E.L. Velasquez 625

TODA LATTICES ON FINITE SYMMETRIC SPACE GRAPHS

A. Jorba, J. Villanueva 628EFFECTIVE STABILITY AROUND PERIODIC ORBITSOF THE SPATIAL RTBP

M.N. Vrahatis, T.C. Bountis, M. Kollmann

ON THE COMPUTATION OF PERIODIC ORBITS ANDINVARIANT SURFACES OF 4D-SYMPLECTIC MAPPINGS

633

M.N. Vrahatis, O. Ragos, F.A. Zafiropoulos, E.C. Triantafyllou 638

ON THE COMPUTATION OF ALL THE EQUILIBRIUM POINTSIN HAMILTONIAN SYSTEMS WITH THREE DEGREES OFFREEDOM

M.N. Vrahatis, O. Ragos, G.S. Androulakis 642

A METHOD FOR COMPUTING FAMILIES OF PERIODICORBITS BASED ON UNCONSTRAINED OPTIMIZATION

H. Wadi 646

PHASE SPACE GEOMETRY OF REACTIVE SCATTERING

R.L. Warnock, J .S. Berg 649

NUMERICAL CONSTRUCTION OF THE POINCARE MAP,WITH APPLICATION TO ACCELERATORS

List of authors 654

Subject index 655

PREFACE

This volume contains the Proceedings of the NATO Advanced Study Institute held inS'Agar6, a small resort in the Costa Brava, Catalonia (Spain), between June 19 and June30, 1995, under the title Hamiltonian Systems with Three or More Degrees of Freedom.

The Institute presented a rather complete survey of what was known in Hamilto-nian systems with 3 and more degrees of freedom and related topics. Hamiltonian sys-tems appearing in most of the applications are non integrable. Hence methods to provenon-integrability results were presented and the different meanings attributed to non-integrability were discussed. For systems near an integrable one, one can show, undersuitable conditions, that some part of the integrable structure, most of the invariant tori,survives. A revision of the KAM theorem and effective methods for the construction ofinvariant tori were presented. Also near integrability averaging methods can be used andthere is a strong relation with the exponentially small phenomena and the adiabatic in-variance. Long time stability of Hamiltonian Systems close to integrable was anotherdiscussed characteristics, as well as diffusion mechanisms (ollowing a variational approach.Some methods to describe the global phase space structure and the related dynamics werealso presented. Different classes of phenomena showing regular and impredictible regionsof motion were shown. Procedures to test the quasiperiodicity were revisited, such asfrequency analysis.

From a topological point of view some singularities must appear in different problems,either caustics, geodesics, moving wavefronts, etc. This is also related to singularities inthe projections of invariant objects, and can be used as a "signature" of these objects.Hyperbolic dynamics appears as a source of impredictible behavior and several mechanismsof hyperbolicity were presented. The destruction of tori leads to Aubry-Mather objects,and this was also touched for a related class of systems. Examples without periodic orbitswere constructed, against a classical conjecture.

Other topics concerned higher dimensional systems, either finite (networks and local-ized vibrations on them) or infinite, like the quasiperidic Schrodinger operator or nonlinearhyperbolic PDE displaying quasiperiodic solutions.

Most of the applications presented in the lectures concerned celestial mechanics prob-lems, as the asteroid problem, the design of spacecraft orbits or methods to computeperiodic solutions.

Eleven morning sessions and part of two afternoon sessions were devoted to lectures.The rest of the afternoons were devoted to 10 seminars, conducted by the lecturers, on thetopics: Structure of the phase space and global dynamics, Bifurcations, Invariant tori andinvariant manifolds, Splitting of separatrices, Numeric and symbolic tools for Hamiltoniansystems, Applications to celestial mechanics, space science, plasma physics, accelerators,etc, Passage through resonance, Detection and measure of the non integrability, Diffusion:geometry and estimates and Adiabatic invariants. Furthermore three round tables weredevoted to integrability, exponentially small phenomena and diffusion. Lots of applications

xv

xvi

showed up during the seminars. Just to mention a few: Molecular dynamics, scattering,electron plasmas, motion of natural and artificial celestial bodies, gyroscopic motions,transport phenomena, atomic systems, hadron colliders, etc. Some poster sessions helpedthe students to present their results.

I consider that both from the scientific and the human points of view the ASI was asuccess and that it will be extremely useful as a reference for future research in that field.It was a unique opportunity to concentrate the efforts of many persons on the background,concrete exemples and methodology of this area of research. One should also stress thatthe ASI provided an excellent opportunity for the contact between researchers in similartopics. An important part of the "free time" was devoted to informal discussions andfruitful exchanges of ideas. At the moment of publication of these Proceedings I am awareof a good amount of collaborations that were started during this AS1.

Beyond 15 lecturers and 80 students, the ASI attracted a strong interest. Nine addi-tional lecturers were present as invited /!;uests, and up to 25 more persons were present inthe sessions as ob~ervers. Many more researchers applied to participate in this NATO ASIthan could be accomodated.

The Organizing COlllmittee (Professors C. Simo (Director), A. Delshallls. R. de laLlave, A. Neishtadt and 1. Seimenis) would like to take this opportunity to thank all thosewhose contributions made the ASI a sllccess. Beyond the key support from NATO, thespanish Ministery of Education, the regional authorities and the Universities of Barcelonaand Poly technical of Catalonia are acknowledged. The staff of S'Agara Hotel provided anexcellent frame for a pleasant work. I would like to thank the colleagues and students oft.he above mentioned Universities who helped liS in the organizing tasks, wit.h a specialmention to the strong support. of Allladeu Delshatlls.

The publicat.ion of these Proceedings has been delayed beyond reasonable limits. Thisis my responsibility and I apologize for it., thanking all the participants and Kluwer stafffor the patience. The unvaluable support of Amadeu Delshams, Angel Jorba and Rafaelde la Llave was essential in the completion of the editorial tasks.

As a sad note we have to regret the unexpected decease of two participants: MikhailTabanov in August 96 and Michele Moons in January 1998. We all remember their enthu-siasm and active participation at the ASI and they will remain in our memory.

Carles SimaDept. de Matematica Aplicada i AnalisiUniversitat de BarcelonaBarcelona, Spain

List of Participants

Simonetta AbendaDepartment of Mathematics, University of Bo-logna, Piazza Porta San Donato, 5, 1-40127Bologna BO, Italyabenda@ciram.ing.unibo.it

George AndroulakisDepartment of Mathematics, University of Pa-tras, Patras 26110, Greecevrahatis@math.upatras.gr

Sigurd AngenentMathematics Department, University of Wis-consin, Van Vleck Hall, 480 Lincoln Drive,Madison, WI 53706, USAangenent@math.wisc.edu

Mickael AntoniINFM, Laboratorio FORUM, Universita diFirenze, Largo Fermi, 2, 1-50139 Firenze, Italyantoni@utp.imt-mrs.fr

Vladimir 1. ArnoldCEREMADE Universite Paris-Dauphine, P.de Marechal de LaUre de Tassigny, 75775Paris 16, France & Steklov Mathematical In-stitute, Moscowarnold@paris9.dauphine.fr

Keith M. AtkinsDepartment of Chemistry, University of Dur-ham, South Road, Durham, DHI 3LE, Eng-landK.M.Atkins@durham.ac.uk

Claude BaesensDepartment of Applied Mathematics andTheoretical Physics (DAMTP), University ofCambridge, Silver St, Cambridge CB3 9EW,U.I

xviii

George ContopoulosDepartment of Astronomy, University ofAthens, Panepistimiopolis, GR-15784 Athens,GreecegcontopQatlas.uoa.ariadne-t.gr

Jacky Cressonlnstitut de Mathematiques de Jussieu, Univer-site Paris 6, UMR 9994, 4 place Jussieu, 75252Paris Cedex 05, FrancecressonQmath.jussieu.fr

Amadeu DelshamsDepartament de Matematica Aplicada I, Uni-versitat Politecnica de Catalunya, Diagonal647, E-08028 Barcelona, Spainamadeu~mal.upc.es

Sergei DovbyshInstitute for Mechanics, Moscow State Univer-sity, Michurinski prot, d.l, 117192 Moscow,RussiadovbyshQrnech.rnath.rnsu.su

Holger R. DullinInst.itut fur Theoretische Physik, UniversiUit.Bremen, Postfach 330440, 28344 Bremen,Germanyhdullin~co]orado.edu

H. Scott Duma.~Department of Mathcmatics, University ofCincinnati, Cincinnati, OH 45221-0025, USAdumasQrnath.uc.edu

Christos Eft.hymiopoulosDepartment. of Ast.ronomy, University ofAthens, Panepistimiopolis, GR-15784 Athens,Greececefthirn@atlas.uoa.ariadne-t.gr

1. Hakan EliassonDepartment of Mathematics, Royal lnst.itut.eof Technology, S-10044 St.ockholm, Swedenhakane~rnath.kth.se

Thomas FabriceDepartement CERGA, OCA-Observatoire deNice, B.P. 229, 06304 Nice Cedex 4, FrancethomasQobs-nice.fr

Corrado FalcoliniDipartimento di Matematica, Universit.a diRoma "Tor Vergata", Via della Ricerca Sci-entifica, 00133 Roma, Italyfalco~rnatrrn3.mat.uniroma3.it

Yuri N. FedorovDepartment of Mathematics and Mechanics,Moscow State University, Moscow 119899,Vorob'evy Gory, Russiafedorov~nw.math.msu.su

Sylvio Ferraz-MelloInst.it.uto Astroniirnico e Geofisico, Universi-dade de Sao Paulo, Caixa Postal 9638, 01065-Sao Paulo, SP, Brasilsylvio~iagusp.usp.br

Elisabeth FerrerDepartament de Matematica Aplicada i Ana-lisi, Universitat. de Barcelona, Gran Via, 585,08007 Barcelona, Spaineferrer@maia.ub.es

Sebastian FerrerGrupo de Mecanica Espacial, Universidad deZaragoza, 50009 Zaragoza, Spainferrer~msf.unizar.es

Joaquim FontDepartament de Matcmatica Aplicada i Ana-lisi, Universitat de Barcelona, Gran Via, 585,08007 Barcelona, Spainquim@maia.ub.es

ErIlest FontichDepartament de Matematica Aplicada i Ana-lisi, Universitat de Barcelona, Gran Via, 585,08007 Barcelona, Spainfontich@cerber.rnat.ub.es

Giovanni GallavottiDipartimento di Fisica, I Universita di Roma,pzza Aldo Moro 2, 00185 Roma, Italygiovanni~ipparco.romal.infn.it

Vassili GelfreichInstitut fUr Mathematik I, Freie UniversitatBerlin, Arnimallee 2-6, D-14195 Berlin, Ger-manygelf~math.fu-berlin.de

Guido GentileIHES, 35 Route de Chartres, 91440 Bures surYvette, Francegentileg~ipparco.romal.infn.it

Antonio GiorgilliDipartimento di Matematica, Universita diMilano, Via Saldini 50, 20133 Milano, Italyantonio~mi.infn.it

Chris GoleDepartment of Mathematics, Smith College,Burton Hall, Smith College, orthamptonMA 01060, USAcgole~math.smith.edu

Gerard GomezDepartament de Matematica Aplicada i Ana-lisi, Universitat de Barcelona, Gran Via, 585,08007 Barcelona, Spaingomez~cerber.mat.ub.es

Angela GrauDepartament de Matematica Aplicada II, Uni-versitat Politecnica de Catalunya, Pau Gar-gallo 5, E-08028 Barcelona, Spainangela~ma2.upc.es

Miquel GrauDepartament de Matematica Aplicada II, Uni-versitat Politecnica de Catalunya, Pau Gar-galle 5, E-08028 Barcelona, Spaingrau~ma2.upc.es

xix

Clodoaldo Grotta Ra.gazzoIME-Universidade de Sao Paulo, CP 66281,05315-970, Sao Paulo, SP, Brasilragazzo~ime.usp.br

Faruk GiingorDepartment of Mathematics, Faculty of Sci-ence, Istanbul Technical University, ITU,80626, Maslak-Istanbul, Turkeyf egungor~cc. i tu. e,du. tr

Pere GutierrezDepartament de Mal.ematica Aplicada II, Uni-versitat Politecnica de Catalunya, Pau Gar-gallo 5, E-08028 Barcelona, Spaingutierrez~ma2.upc.es

George HallerDivision of Applied ~\1athematics, Brown Uni-versity, Providence, RI 02912, USAhaller~cfm.brown.edu

Heinz Hanl3mannRijksuniversiteit Groningen, Vakgroep Wis-kunde, Postbus 800, 9700 AV Groningen, TheNetherlandsHeinz~iram.rwth-aachen.de

Alex HaroDepartament de Matematica Aplicada i Ana-lisi, Universitat de Barcelona, Gran Via, 585,08007 Barcelona, Spainharo~cerber.mat.llb.es

Jacques HenrardFUNDP, Departement de Mathematique, 8,Rempart de la Vierge, B-5000 Namur, Bel-giumjhenrard~math.fundp.ac.be

Michel R. HermanDepartement de Mathematique, Ecole Poly-technique, 91128 Palaiseau, France

xx

Poul G. HjorthDepartment of Mathematics B-303, Techni-cal University of Denmark, DK-2800 Lyngby,Denmarkhjorth~rnat.dtu.dk

Simos IchtiaroglouDepartment of Physics, University of Thessa-loniki, 54006, Greeceichtiaroglou~olyrnp.ccf.auth.gr

Maylis IrigoyenUniversite Paris 2, 92 Rue d'Assas, 75006Parisirigoyen~rnath.jussieu.fr

Charles JaffeDepartment of Chemistry, West Virginia Uni-versity, Morgantown, WV 26506-6045 USAuOd96~wvnvrn.wvnet.edu

Christopher JarzynskiInstitute for Nuclear Theory, NK-12, Univer-sity of Washington, Seattle, WA 98195, USAchrisj@rnocha.phys.washington.edu

Hans-Rudolf JauslinLaboratoire de Physique, Cniversite de Bour-gogne, B.P. 138 - Fac. Sciences Mirande, F-21004 Dijon cedex, Francejauslin~satie.u-bourgogne.fr

Angel JorbaDepartament de Matematica Aplicada i Ana-lisi, Universitat de Barcelona, Gran Via, 585,08007 Barcelona, Spainangel~rnaia.ub.es

Srihari KeshavamurthyDepartment of Chemistry, Baker Laboratory,Cornell University, Ithaca, NY 14853, USAsrihari@chernres.chern.comell.edu

Tetsuro KonishiR-Lab., Department of Physics, School ofScience, Nagoya University, Nagoya,464-01,Japantkonishi~allegro.phys.nagoya-u.ac.jp

Igor Y. KostyukovPlasma Physics and High-Power Electron-ics, Institute of Applied Physics, RussianAcademy of Science, 46, Uljanov Street,603600, izhny Novgorod, Russiakost@appl.sci-nnov.ru

Valery V. KozlovDepartlllent of Mathematics and lVIechanics,Moscow State University, Moscow 119899,Vorob'evy Gory, Russiakozlov@nw.rnath.rnsu.su

Jacques LaskarCNRS-Astronomie ct Systemes D.vnarniques,Bureau des Longitudes, 3 rue Mazarine, 75006Paris, Francelaskar@bdl.fr

.J. Tomas LizaroDcpartament de Matematica Aplicada i Tele-matica, Universitat Politecnica de Catalunya,Victor Balaguer sin, E-08800, Vilanova i laGeltni, Barcelona, Spainlazaro@rnat.upc.es

Lev. j\/. LerrnanResearch Institute for Applied Mathemat.ics &Cybernet.ics, 10 UI'yanov St.., Nizhny ov-gorod, 603005, Russialerrnan~focus.nnov.su

Rafael de la L1aveDepartment of Mathematics, University ofTexas at Aust.in, Austin, TX 78712, USAllave@rnath.utexas.edu

Martin \V. Lo.Jet Propulsion Laborat.ory, California Inst.i-tute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USArnwl@trantor.jpl.nasa.gov

Pierre LochakDepartement de Mathematiques, Ecole Nor-male Superieur, 45 rue d'Ulm, 75230 ParisCedex 05Pierre.Lochak@ens.fr

Andrzej J. MaciejewskiInstitute of Astronomy, Nicolaus CopernicusUniversity, 87-100 Torun, Chopina 12/18,Polandmaciejka@astri.uni.torun.pl

Robert S. MacKayDepartment of Applied l\Iathematics andTheoretical Physics (DAMTP), University ofCambridge, Silver St, Cambridge CB3 !.JEW,UKR.S.MacKay@damtp.cam.ac.uk

Jean-Pierre MarcoInstitut de Mathemat.iques de Jussieu, Univer-site Paris 6, UMR 9994, 4 place Jussieu, 75252Paris Cedex 05, Francemarco@math.jussieu.fr

Pau MartinDepartament de Mat.ematica Aplicada i Tcle-matica, Universit.at. Politccnica de Cat.alunya,Gran Capita s.n., Madul C3, 08034 Barcelona,Spainmartin@mat.upc.es

Jose Martinez-AlfaroDepart.ament. de Matematica Aplicada, Uni-versit.at de Valencia, Dr. MolineI' 50, 46100Burjassot., Valencia, Spainmartinja@uv.es

Josep MasdemontDepart.ament de Matemat.ica Aplicada I, Un i-versitat. Politecnica de Catalunya, Diagonal647, E-08028 Barcelona, Spainjosep@barquins.upc.es

Vieri MastropietroDipartimento di Matematica, II Universit.a diRoma, 00133 Roma, Italyvieri@ipparco.romal.infn.it

xxi

Mikhail V. MatveyevDepart.ment of Different.ial Equat.ions, Mos-cow Aviation Inst.it.ute, Volokolamskoe shosse4, 125871 Moscow, Russiamatveyev@k804.mainet.msk.su

James D. MeissProgram in Applied Mat.hematics, Universityof Colorado, Boulder, CO 80309-0526, USAjdm@boulder.colorado.edu

Jan-Robert. von MikzewskiMax-Planck-Inst.it.ut fUr Astrophysik, Karl-Schwarzschild-St.raf3e 1, 85740 Garching, Ger-manyjrm@mpa-garching mpg.de

Arnau Mil'Depart.ament. de Matemat.iques i Inforrnat.ica,Universit.at. de les Illes Balears, Crt.a. deValldemossa, Km. 7,5, 07071 Palma, SpaindmiamtO@ps.uib.es

Michele MoonsFUNDP, Departement. de Mathcmatique, 8,Rempart. de la Vierge, 8-5000 Namur, Bel-giummmoons@quick.cc.:fundp.ac.be

Juan J. MoralesDepart.ament de Matemat.ica Aplicada II, L'ni-versit.at. Politecnica de Cat.alunya, Pau Gar-gallo 5, E-08028 Barcelona, Spainmorales@ma2.upc.es

Alessandro MorbidelliC.N.R.S., Observatoire de la Cot.e d'Azur.B.P. 229, 06304 Nice Cedex 4, Francemorby@obs-nice.fr

Jurgen MoserMathematics ETH-Zurich, ETH-ZentrumCH-8092 Zurich, Swit.zerlandmoser@math.ethz.ch

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Anatoly I. NeishtadtSpace Research Institute, Profsoyuznaya84/32, Moscow 117810, Russiaaneishta~mx.iki.rssi.ru

Laurent NiedermanTopologie et Dynamique, Universite ParisSud, U.R.A. 1169, Batiment 425 (Mathema-tiques) 91405 Orsay, Francelaurent~bdl.fr

Ana NunesUniversity of Lisbon, Campo Grande, ED Cl,piso 4, 1700 Lisbon, Portugalanunes~ptmat.lme.fe.ul.pt

Merce OUeDepartament de Matematica Aplicada 1, Uni-versitat Politecnica de Catalunya, Diagonal647, E-08028 Barcelona, Spainolle~mal . upe. es

Yakup OzkazancDepartment of Electrical Engineering, Hacet-tepe University, Beytepe, Ankara, Turkeyyakup~leo.ee.hun.edu.tr

Yannis PapaphilippouCNRS-Astronornie et Systernes Dynamiqucs,Bureau des Longitudes, 3 rue Mazarine, 75006Paris, Franceyannis~mail.eern.eh

Nuno S. Andrade PereiraFaculty of Sciences, University of Lisbon, Av.Almirante Reis no. 173, 20 Esq. 1000, Lisboa,Portugalnpereira~eosmos.ee.fc.ul.pt

Jiirgen P6schelDepartment of Mathematics. UniversitatStuttgart, Stuttgart, Germanyposehel~mathematik.uni-stuttgart.de

Rafael Ramirez-InostrozaDepartament d'Enginyeria Informatica, Uni-versitat Rovira i Virgili, Crta. de Salou, sin,43006 Tarragona, Spainrramirez~etse.urv.es

Rafael Ramirez-RosDepartament de Matematica Aplicada I, Uni-versitat Politecnica de Catalunya, Diagonal647, E-08028 Barcelona, Spainrafael~tere.upe.es

Vanessa RobinsDepartment of Theoretical Physics, ResearchSchool of Physical Sciences, Australia Na-tional University, Canberra ACT 0200, Aus-traliavbrl05rsphyl.anu.edu.au

Philippe RobutelCentre de Recerca Matematica, Apartat 50,E-08193 ReUaterra, Barcelona, Spainrobutelbdl.fr

Vered Rom-KedarThe Department of Applied Mathematics andComputer Science, The Weizmann Institute ofScience, P.O.B. 26, Rehovot 76100, Israelvered~wisdom.weizmann.ae.il

Vassilios RothosDepartment of iVlathematics, Uni versity of Pa-tras, Patras 26110, Greecerothosmath.upatras.gr

Michael RudnevDepartment of Mathematics, University ofTexas at Austin, Austin, TX 78712, USAmrumath.utexas.edu

David SauzinCNRS-Astronomie et SystEmIes Dynamiques,Bureau des Longitudes, 3 rue Mazarine, 75006Paris, Francesauzin~bdl.fr

Luca SbanoSISSA/ISAS-International School for Advan-ced Studies, via Beirut 2/4, 34014 Trieste,Italysbano~neumann.sissa.it

Daniel J. ScheeresJet. Propulsion Laboratory, California Insti-tute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USAseheeres~iastate.edu

George W. SchlossnagleDepartament of Mathematics, Brown Univer-sity, 17 Forest St, Providence, RI 02906, USAgeorge~gauss.math.brown.edu

Constance SchoberProgram in Applied Mathematics, Universityof Colorado, Boulder, CO 80309, USAsehober~math.odu.edu

Christian G. SchroerInstitut fUr Fest.korperforschung, Forschungs-zentrum Jiilich, D-52425 Jiilich, Germanyeh.sehroer~kfa-juelich.de

Tere M. SearaDepartament de Matematica Aplicada I, Uni-versitat Politecnica de Catalunya, Diagonal647, E-08028 Barcelona, Spaintere~mal.upc.es

Iannis SeimenisDepartment of Mathematics, University of theAegean, Samos 83200, Greeceisei~ru.ath.aegean.gr

Mikhail B. SevryukInstitute of Energy Problems of ChemicalPhysics, Lenin prospect 38, Bldg. 2, Moscow117829, Russiarusin~iepcp.msk.su

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Dmitry SharomovDepartment of Mathematic;:al Physics, PhysicsFaculty, Sankt-Petersburg State University,198904, Sankt-Petersburg, St. Petergof, P.O.Box 142, Russiasharomov~misha.usr.saai.ru

Vladislav V. SidorenkoKeldysh Institute of Applied Mathematics,Miusskaja sq. 4, Moscow, 125047, Russiasidorenk~applmat.msk.su

Carles SimaDepartament de Matematica Aplicada i Ana-lisi, Universitat de Barcelona, Gran Via, 585,08007 Barcelona, Spaincarles~maia.ub.es

Ya. G. SinaiDepartment of Mathematics, Princeton Uni-versity, Princeton, New Jersey 08544, USAsinai~math.princeton.edu

Charalambos SkokosDepartment of Astronomy, University ofAthens, Panepistirniopolis, GR-15784 Athens,Greecehskokos~atlas.uoa.ariadne-t.gr

Daniel SteichenCNRS-Astronomie et Systernes Dynamiques,Bureau des Longitudes, 3 rue Mazarine, 75006Paris, Francesteichen~bdl.fr

Teresa J. StuchiInstituto de Fisica, Universidade Federal doRio de Janeiro, Rio de Janeiro, Braziltstuchi~if.ufrj.br

Toni SusinDepartament de Matematica Aplicada I, Uni-versitat Politecnic

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Eduardo TabacmallThe Geometry Center, University of Min-nesota, 415 Lind Hall, 207 Church St. SE,Minneapolis, MN 55455, USAtabacman~geom.umn.edu

Mikhail B. TabanovChair of Applied Mathematics, St.-PetersburgState Academy of Aerospace Instrumenta-tions, Bolshaja Morskaja 67, St.-Petersburg,190000, Russiatabanov~misha.usr.saai.ru

Joan Carles TatjerDepartament de Matematica Aplicada i Ana-lisi, Universitat de Barcelona, Gran Via, 585,08007 Barcelona, Spainjcarles~maia.ub.es

Ezio TodescoINFN, University of Bologna, Via Irnerio, 46,1-40126 Bologna BO, Italyetodesco@bo.infn.it

Stathis TompaidisIRMAR, Ulliversite de Relllles L 35042Rennes Cedex, Francestathis~athena.bus.utexas.edu

Alexander TovbisDepartment of Mathematics, West VirgilliaUniversity. Morgantown, \VV 26506-6::>10USAatovbis~pegasus.cc.ucf.edu

Omitry TreschevDepartment of Mathematics and Mechanics,Moscow State University, Moscow 119899.Vorob'evy Gory, Russiadtresch@mech.math.msu.su

Kell UmenoFrontier Research Program, The Institute ofPhysical and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama 351-01, Japanchaosken@giraffe.riken.go.jp

Turgay UzerSchool of Physics, Georgia Institute of Tech-nology, Atlanta, Georgia 30332-0430, USAturgay.uzer@physics.gatech.edu

Alexei A. VasilievSpace Research Institute, Profsoyuznaya84/32, Moscow 117810, Russiaalex@valex.iki.rssi.ru

Elillor VelasquezMathematics Department, University of Cali-fornia, Berkeley, CA 94720-3840 USAvelasque@math.berkeley.edu

Jordi VillanuevaDepartament de Matematica Aplicada I, Uni-versitat Politecnica de Catalunya, Diagonal647, E-08028 Barcelona, Spainjordi@tere.upc.es

Michael 1\. VrahatisDepartment of Mathematics, University of Pa-tras, Patras 26110, Greecevrahatis@math.upatras.gr

Hichalll Iv!. WadiLaboratoire de Spectrometric Physique, Uni-versite Joseph-fourier-Grenoble, BP 87 F-38402 Saint-Martin-d'Heres cedex, FranceHicham.WADI~ujf-grenoble.fr

Robert L. \VarnockStanford Linear Accelerator Center, StanfordUniversity, Stanford, C'.\ 94309, USAwarnock@mailbox.slac.stanford.edu

Stephen WigginsControl and Dynamical Systems, 104-44 Cal-tech, Pasadena CA 91125, USAwiggins@cds.caltech.edu

Haruo Yoshida:'-Jational Astronomical Observatory, Mitaka,Tokio 181, Japanyoshida@cl.mtk.nao.ac.jp

Lectures

INFLECTION POINTS, EXTATIC POINTSAND CURVE SHORTENING

S. ANGENENTMathematics Department, University of WisconsinVan Vleck Hall, 480 Lincoln Dl'ive, Madison, WI 53'106, USA

If (M 2 ,g) is a surface with Riemannian metric, then a family of immersed curves{Ct 10 'S t < T} on M 2 evolves by Curve Shortening if

BC _at = Kgl/, (1 )where Kg is the geodesic curvature, and iJ is a unit normal to the curve. Since KgiJ can bewritten as ~, where s is arclength along C, (1) is essentially a parabolic equation, i.e. anonlinear heat equation.

In [2] it is shown that for any solution {Ct I 0 'S t < T} of (1) there is only a discreteset of times at which the immersed curve Ct will have self-tangendes. Hence the numberof self-intersections of Ct is always finite, and it was also shown in [2] that this numberdecreases whenever the curve develops a self-tangency.

As the name suggests, Curve Shortening is a gradientflow for the length functionalon the space of immersed curves in the surface M2. One can therefore try to use CurveShortening to prove existence of geodesics by variational methods. In my talk at S'Agar6I observed that geodesics always are curves without self-tangencies, and recalled that thespace of such curves has many different cor.nected components. I then discussed how onecan try to exploit the nice behaviour of ClJr've Shortening with respect to self-intersectionsto prove existence of geodesics in each component.

The fact that Curve Shortening never increases the number of self-intersec+.ions of acurve is a consequence of a theorem of Sturm on linear parabolic equations, and insteadof describing the contents of my talk I would like to point out that this theorem of Sturmcan also be used to give alternative proofs of the following theorems of Arnol'd:

Theorem A. ("Tennis ball theorem") Any embedded curve in 52 which dividesthe sphere into two parts of equal area has at least four inflection points.

Theorem B. Any noncontractible embedded curve in IR IF 2 has at least three inflectionpoints.

Theorem C. Any plane convex curve has at least four inflection points and six extaticpoints.

Proofs of these theorems can be found in [3] and in the preprints made available at theconference in S'Agaro. One of these preprints [4] contains a fourth theorem on "flatteningpoints" of space curves which Arnol'd puts in the same list of generalizations of the Morseinequalities. I have not been able to find a proof of this particular theorem along the samelines as the proofs of theorems A, 8 and C which I will give below.

3

4

The tht-orem Lj St.urIn ,;\tieh we use to give alternative proofs of theorems A, BandC can be :;tated as folluws:

Theorem ([Sturm 1836, [1]]) Let u(x, t) satisfy a linear parabolic PDE of the type

.j fj2 u OU:. --~ a(, t)"'-2 + b(x, t)", + c(x, t)u (2)uL uI uI

[or I E ~, 0 < t < T, and assume thai u(:J.: + I, t) == u(:J.:, t). Assume u, 1Lt, u x , uxx , a,at, ax, axx , b, bx and c are continuous. Assume furtlJennore that the coefficient a(x, t) isstrictly positive.

Then z(t) = #{x E IR/Z I u(x, t) = O} is a finite and nonincreasing [unction of t. Atany time t [or which ue, t) has a multiple zero z(t) will decrease.

At the end of this note I will show how this theorem is :;imilar tv the well known factthat zeroes of a function of one complex variable always have positive degree.

The proof:; of theorems A, Band C will more or less go like this: Given an arbitrarycurve C we consider the maximal evolution {Ct lOS; t < T} of Curve Shortening withinitial data Co = C and determine the asymptotic behaviour of C, as t t T. We then observethat the curvature "'c, satisfies a lincar pilrilholic equation of the form (2), so that Sturm':;theorem tell:; us that C has at least as many inflection points as Ct for t close to T. Fromthe asymptotic behaviour of Ct for t t T we then get the desired lower bound for thenumber of inflection points. To e:;tilllate the number of extatic points we allow the clll'veto evolve by "Affine Curve Shortening" and con:;ider the affine curvature J.l instead of theEuclidean curvature.

1. The Tennis ball Theorem

Let Co C S2 be all embedded curve which divides the sphere in two parts of equal area,and let {Ct lOS; t < T} be the corresponding evolution by Curve Shortening. If oneparametrizes Ct by C : IR/Z x [0, T) S2 with OtC 1- 01:C, then the curvature Ii: (x, t) of C/at C(x, t) satisfies

ali: cj21i: 2at = as2 + (II: + 1)11:, (3)

where a/as = IJxCI~ 10/Dx. Thi:; equation is of the form (2) so that the number of zeroesof ",e, t), i.e. the number of infiection poillts of Ct does not increase with t.

Lemma. The evolution Ct exists for all t > O. Ai any time 1 the curve Ct divides thesphere into two parts o[ equal area. As t --+ oc the curve Ct converges to a great circle.

Denote by O(t) one of the two components of S2 \ Ct and let A(t) be the area of ~l(t).Then

1I'(t) = / vds = / "'ds,JC t JC t

where v is the normal velocity and", is the clll'vature of Ct in the direction of the outwardnormal of Of,. The sphere has Gauss curvature K == +1 so the Gauss-Bonnet theorem tellsus that

A'(t) = / r;ds = -21f + f / Kdll = 11(1) - 21f.Jc, JOtBy our assumption A(O) = 2rr and it thus follows that A(t) == 27l' for all t E [0, T).

By Grayson's theorem [6] the solution Ct either shrinks to a point in finite time or elseexists for all t ::::: O. Since Ct bounds a region 0t of area 27l' it cannot shrink to a point andhence must exist forever.

5

The w-limit set of the evolution Ct consists of geodesics, and since the flow is realanalytic an argument of Leon Simon implies that the Ct converge to a unique geodesic Cooof 52. Such a geodesic must of course be a great circle. To determine how many inflectionpoints Ct has for large t we choose coordinates and linearize Curve Shortening around thelimit Coo.

We may assume that Coo is the equator, i.e. the intersection of 52 with the xy-plane.After removing north and south poles we can then project the sphere onto the cylinderx2 + y2 = 1, which gives us coordinates (, z). In these coordinates the equator is givenby z = and any C 1 nearby curve is the graph of a 27f periodic function z = u(). Forinstance, any great circle which is not a meridian is the graph of u() = A cos( - o) forcertain A E JR, o E JR/27f1Z.

If t is large then Ct will be a graph z = u(, t). Curve Shortening implies that u(, t)satisfies

(4)

where A(u,u) is some smooth positive function with A(O,O) = 1. One can compute Aexplicitly, but the precise form of the equation is not important. Instead we observe thatsince great circles do not evolve under Curve Shortening, the functions u = M cos( - o)must be steady states of (4) for any value of M, o.

Our solution u(, t) together with its derivatives converge to C'J, == as t -t 00. Fromthis one can deduce that u must be asymptotic to a solution of the linearized equationcorresponding to (4), i.e. to

Thus(5)

for some n 2: 2 and some C =I- 0. Here 0(- .. ) represents some function which is small inCk for any k < 00.

The proof is now complete since the graph of u(-, t) will have at least 2n 2: 4 inflectionpoints.

Nearly the same argument gives us the theorem on inflection points of simple noncon-tractible curves in JR 1P'2. Indeed, if /'0 c JR 1P'2 is such a curve then its lift Co to the unitsphere is an embedded curve which divides the sphere into two parts of equal area. Thelift Co is also invariant under the antipodal map x >--+ -x. As above the corresponding evo-lution {Ct I t 2: O} by Curve Shortening will converge to a great circle, with asymptoticsgiven by (5) for some n 2: 2. Since 3,11 Ct must also be invariant under the antipodal map(, z) >--+ ( + 7f, -z) only odd values of n can occur in (5). Hence the lowest value of nwhich can appear is n = 3. For any n 2: 3 the curve with graph Ecos(n( - o)) has atleast 6 inflection points. By Sturm's theorem Co must have at least 6 inflection points, andthe curve /'0 in the projective plane must have at least 3 inflection points.

2. Extatic Points

Let C C JR2 be a convex curve. For any point P E C there will be a conic section

Ax2 + Bxy + C y 2 + Dx + Ey + F = 0

6

which has maximal order of contact with C at P. In general this order of contact will be5. If the order of contact is 6 or more the point P is called extatic. We can also describethese points in terms of affine geometry.

Recall that the affine arc length on a convex curve is defined by

where w is the area or symplectic form on IR2 . If one parametrizes C by affine arclengthone has w(Cs,Css ) == 1, and hence after differentiation w(Cs,Csss ) == O. It follows that forsome J.L E IR one has

The quantity J.L is called the affine curvature of C. Conic sections are exactly those curveswhich have constant affine curvature. One easily verifies the following:

Lemma. P E C is extatic if and only if c::: (P) = O.We must therefore show that the affine curvature has at least 6 critical points on any

convex curve. To do this we let the curve evolve by affine cm've shortening, i.e. with normalvelocity v = (K) 1/3. In terms of a parametrization this is equivalent with the PDE

which is formally similar to (1), but actually is different since here s is affine arclength,a _ { -1/3 aand thus a::; - w(Cx,Cxx )} ax'

Let p( B, t) be the support function of Ct , i.e.

p(O, t) = sup{xcosB + ysinB I (x,y) E Cd

The support function p completely determines the curve Ct, and the curvature, velocityand affine curvature are given by

K = - (p + POO)-l , V = - (p + P8O)-lj3,J.L = v3 (V8O + v) .

Starting from v = rlt one then computes that v(B, t) and J.L(O, t) evolve according to3~ = v" (voo + v) = vp,

(6)

(7)

a3N' = 1)4 pOO + 2V3VoJLO + 4JL2, (8)and hence, after differentiating (8) with respect to B, one finds that J.Lo satisfies a linearparabolic PDE of the type (2). Sturm's theorem therefore says that affine curve shorteningdoes not increase the number of extatic points of a convex curve.

Recall that Sapiro and Tannenbaum [8] showed that the Ct shrink to a point at sometime T > 0, and that the Ct , after rescaling by a factor (T - t) -3/", converge to anellipse. After applying a translation and a special affine transformation we may assumethat the limiting point is the origin, and that the limiting ellipse is a circle. Thus the curve(T - t)3/4Ct converges to a circle, and the rescaled curvature (T ~ t)3/4 K(B, t) and rescaledvelocity (T - t)I/4 v (B, t) converge in C= to constants.

7

We put w(B, T) == (T - t)I/4 V (B, t), and T == -In(T - t), and observe that (7) implies

OW w 4 W- == - (woo + w) - -.OT 3 4 (9)

The constant to which v converges as t t T, i.e. when T -+ 00, must be a time independent

( )1/4

solution of (9), from which one finds that w(B, T) -+ ~ (T -+ (0).

To detect the oscillations of w for large T we linearize (9), i.e. we put w(B, T) == 0r/4 +7/J(B, T), and after discarding higher order terms in 7/J find that 7/J satisfies

(10)

For large values of T any solution of this equation is asymptotically like

(11)

for some kEN, C > 0 and Bo. Since 7/J vanishes as T -+ 00 we must have k 2: 3.If we now substitute (11) back in (6), we get an asymptotic expansion for J-L(B, t),

3 I k 2 /4 2J-L(B, t) ~ 4(T _ t) + C (T - t) - cos k(B - Bo) + ... ,

from which it follows that J-L has at least 2k 2: 6 critical points.To prove the four vertex theorem one evolves a convex curve by (ordinary, Euclidean)

Curve Shortening and applies exactly the same argument. The analog of the Sapiro-Tannenbaum theorem is the result of Gage and Hamilton [5], which says that after rescalingby (T - t) -I /2 the curve converges smoothly to a circle of radius .;2. The resulting proofis of course much more complicated than the textbook proof.

3. Parabolic equations as degenerate Cauchy-Riemann equations.

One can present Sturm's theorem as an analog of a well known fact concerning analyticfunctions: Any nondegenerate zero of an analytic function has positive degree; by a smallperturbation of the function one can decompose a degenerate zero into several nondegen-erate zeroes and conclude that any zero (simple or not) has positive degree.

To see the analogy, consider u(x, t) a smooth function on rl == IRj&: X [0, T] whichsatisfies (2). If u(-, to) has simple zeroes for some to E [0, T], then the number of zeroes ofu(', to) is twice the winding number of the map w : rl -+ IR2

w(x,t) == (:~(x,t),u(x,t))

on the circle IRj&: x {to}. Thus the number of zeroes at time T minus the number of zeroesat time t == 0 is twice the degree of the map w on rl.

Suppose now that w == (v, u) satisfies a first order system of equations of the form

U x == A(x, t)u + B(x, t)v ,

Ut - (3(x, t)vx == C(x, t)u + D(x, t)v,

(12a)

(12b)

8

for certain functions (3, A, B, C, D on n. For instance, this will be true if U satisfies (2)(choose (3(x, t) = a(x, t), A = 0, B = 1, C(x, t) = -c(x, t) and D(x, t) = -b(x, t).)

Positivity Lemma. Assume (3(x, t) > 0 on n. If w is non ;oem on an and if wanlyhas simple zeroes in n, i.e. if detDw i= 0 at any zero of w, then the degree of w : n --t IR2is nonnegative.

The reason is simple: at any zero one has u = v = 0, so that (12a), (12b) imply t.hatU x = 0, and Ut = (3(x, t)vx . Hence

detDw(:J;, t) = I:: ~: I= Iva (3v~x I= (3(:1:, t) (vJ,)2 2: o.If the determinant is non zero, it must therefore be positive.

If the system (12a), (l2b) is such that any solution can be approximated by a solutionwith only simple zeroes (or even a solution of a system of the same type with simplezeroes) then one can drop the condition that w must have simple zeroes: any solution of(12a), (12b) which is nonzero on an must have nonnegative degree.

The system (12a), (12b) is similar to the Cauchy-Riemann equations. Indeed, the samearguments can (and have of course been) applied to equations of the form

U r + ex(x, t)Vt = A(x, I.)u + B(x, t)v ,

"11./ - (3(x, t)vx = C(:E, t)u + D(x, t)v,

(13a)

(13b)

with positive coefficients (x, (3, to yield the same conclusion (so, for ex = (3 = 1 andA = B = C = D = 0 one finds that an analytic function v + iu of a complex variablex + it always has nonnegative degree.)

One obtains the system (12a), (12b) which contains the linear parabolic equation (2)as a special case by letting the coefficient ex tend to zero.

4. Which equations satisfy the positivity lemma?

We will show that up to linear transformations the only systems of t.wo equations in twofunctions of two variables for which the proof of the positivity lemma works, are theCauchy-Riemann equations and the Heat Equation written as a system.

Consider a system of f. PDEs

n

Mi(U) = L n{(x)uj(x);=\

(14)

in n functions UI, ... , Un of n variables X\, ... , Xn, wllere M is a first order differentialoperator

Assume that the degree of any nondegenerate zero of any solution of (14) is positive, forthe reasons given in the proof of the positivity lemma. In other words, we assume thatany nonsingular matrix A jk with MrA jk = 0 actually ha~ detA > O. What can we sayabout the differential operator M?

One can transform the equations (14) in three different ways: one can make linearcombinations of the equations, i.e. one can replace the differential operators Mi by M: =

9

SijMj for an arbitrary S E GL(n, ~). The new and old equations will have the samesolutions, so the positivity lemma will hold for one if and only if it holds for the other. Upto substitutions of this kind, the operators Mi are completely determined by their kernel,i.e. the subspace

x = {A E 'cn I L Mr AJk = OVi}j,k

of the space of all n x n matrices 'cnThus we can rephrase our question as follows: for which subspac:es X C 'cn is detA ~ a

for all A E X?Since det(-A) = (-1) n detA, the dimension n must be even. Examples of such sub-

spaces X C 'cn in any even dimension are given by the kernel of the Cauchy-Riemannequations.

The two other types of transformations which one can apply to the equations (14)are linear transformations of the dependent variables u: = SijUJ , a.nd of the independentvariables x: = RijXj' The corresponding action on 'cn is given by A H S . A . R- 1 If werequire detS, detR > 0, t.hen t.he determinant will be nonnegat.ive on X if and only if it isnonnegative on SXR- 1.

In two dimensions one can now easily classify all equations of the type (14) for whichthe positivity lemma applies.

Let X C 'c2 be a linear subspace on which the determinant is nonnegative. If X is onedimensional, t.hen it is spanned by a matrix with nonnegative determinant. Conversely,any subspace spanned by such a matrix has det~ 0. This situation corresponds t.o a systemof three equat.ions.

Assume that. X is two dimensional. If X only contains singular matrices, then up t.o

linear transformat.ions it. must. be the subspace spanned by (6 ~) and (~6)'If X cont.ains at least one nonsingular matrix A then we may assume that A is t.he

identity matrix after replacing X with A-I X. Being two dimensional, X is spanned by Iand some other mat.rix B, i.e. X = {AI + pB I )." P E ~}. After replacing X with SXS- Ifor suitable S we may assume that B is in Jordan normal form. By subtracting a suit.ablemultiple of I from B we may assume that B has trace 7,ero. Let ,'3 be t.he eigenvalues ofB. Then by assumption

det(AI + f-LB) = ()., + f-L(3)()., - p(3) ~ for all )." ~l E ~. This can only happen if (3 = iw, w ~ is imaginary or zero.

If (3 of. a we find that X consists of all matrices of the form (."b ~), and that thecorresponding differential operator M is equivalent to the Cauchy-Riemann equations. If(3 = 0, then B must be the matrix (~6)' and X consists of all matrices of the form (~~).the corresponding equations are then equivalent to the system (12a), (12b) relat.ed t.o t.heheat equation.

For a geometric view of t.he preceding argument one should identify 'c2 = {( ~ ~) I

a, b, c, d E ~} wit.h ~4, and identify the linear subspace X wit.h the corresponding linearsubspace K C lR lP'3. The equation det. ~ 0, i.e. ad - be ~ defines a subset. of lR lP'3whose boundary is a quadric Q. The question is t.herefore whieh subspaces K C lR lP'3lie on one side of the quadric Q? We found: points (X one-dimensional), lines disjointfrom Q (Cauchy-Riemann equat.ions), lines tangent to but not. contained in Q (equat.ions

10

of parabolic type) and lines contained In Q (X only contains singular matrices and thepositivity lemma is vacuous).

Acknowledgements

I am supported by NSF through grant DMS 9058492. For the academic year my addressis Departamento de Matematica Aplicada, Universidad Complutense de Madrid, Spain,and I receive support from the spanish Ministry of Education in the form of a "beca desabatico."

References

I. S.B.Angenent, The zeroset of a solution of a parabolic equation, J.reine u.angewandte Mathematik,390 (1988) 79-96.

2. S.B.Angenent, Parabolic Equations for Curves on Surfaces If, Ann. of Math. 133 (1991), 171-215.3. V.I.Arnol'd, Topological Invariants of Plane Curves and Caustics, A.M.S. University Lecture Series 5

(1994).4. V.I.Arnol'd, On the numbeT' of fiattening points of space curves, Report No.1, 1994/1995 Institut

Mittag-Leffler, to appear in Advances in Soviet Mathematics (1995).5. M.Grayson & R.S.Hamilton, The heat equation shrinking plane convex curves, J. of Diff. Geom., 23

(1986) 69-96.6. M.A.Grayson, Shortening embedded curves, Annals of Math., 129 (1989) 7l-111.7. C.Sturm, Memoire sur une classe d'equations a differences partielies, J.Math.Pures at Appl. 1 (1836)

373-444.8. G.Sapiro & A.Tannenbaum, On Affine plane Curve ShoTtening, J.Funct.Anal. 119 (1994) 79-120.

TOPOLOGICALLY NECESSARY SINGULARITIES ON MOVINGWAVEFRONTS AND CAUSTICS

VLADIMIR I. ARNOLDSteklov Mathematical Institute, MoscowCEREMADE UniversiU Paris-Dauphine, France

Extended abstract

In his posthumous Vorlesungen Uber Dynamik Jacobi claimed that the caustic (formedby the conjugate points of a given initial point along all the geodesics) on the surface ofan ellipsoid has four cusps.

I do not know whether this statement (which might be called "The Last GeometricalTheorem of Jacobi") is true, but the existence of AT LEAST FOUR cusps on the causticsof the surfaces seems to be a deep topological property of the Hamiltonian systems withmany degrees of freedom.

The corresponding general theorems in symplectic and contact topology contain, alongwith the existence of the four cusps on each caustic for any surface which is sufficientlyclose to the standard sphere, the generalizations of the four vertices theorem of the Eu-clidean plane topology and of the Moebius theorem on the three inflection points on anO:1contractable embedded circle of the projective plane.

Technically speaking, these results are closely related to the Kellogg-Tabachnikov the-orem in Sturm theory, stating that a Fourier series has at least as much zeros, as thelowest harmonics presented in the series. This Sturm theory fact is a generalization of theMorse inequality for the circle (to which it is reduced in the simplest case where the firstharmonic is present).

The Morse theory implies minorations on the number of the periodic orbits and on thenumber of fixed points of the exact symplectomorphisms, which are not too far from theidentity. In 1965 I have suggested that this minoration might hold for large perturbationsas well.

This extension of the Poincare's Last Geometrical Theorem had been later proved (formany different cases) in a series of brilliant works of Conley and Zehnder, Chaperon, Floer,Laudenbach and Sikorav, Gromov, Givental, Ono and many others.

The resulting theory of Lagrangian intersections and Floer homology is a far-reachinggeneralization of the Morse theory to the case of multivalued functions.

The geometrical problems discussed above suggest that the Sturm theory as well asthe Morse theory has a generalization to the multivalued functions case (corresponding tothe arbitrary large perturbations of the "integrable" reversion of a spherical wavefront).

They also suggest highly nontrivial generalizations of the Sturm theory to the case ofthe functions of more than one variable, leading to many conjectures on the integrableand nonintegrable dynamical systems, on the Legendrian knots, Lagrangian singularities

II

12

and on the elementary (but very difficult) topology of the plane curves (which might beconsidered as the noncommutative version of the knot theory, which is, from this point ofview, a simplified, commutative version of the theory of the plane curves).

References

1. Arnold, V.I. (1994) Topological Invariants of plane curves and caustics, AMS, Univ Leelur'es, vol. 5,Providence, 60 pp.

2. Arnold, V.I. (1994) Plane curves, their invariants, perestroika.~ and bifurcations, in Singularities andbifurcations, Advances in Soviel Mathematics, Vol. 21, AMS, Providence, pp. 33-91.

3. Arnold, V.I. (1993) Sur les proprietes topologiques des projections Lagrangiennes en geometrie sym-plectique des caustiques, Cahiers de Mathematiques de la decision, n. 9320, CEREMADE, UniversiteParis-Dauphine, 14/6/93, 9 pp. (also in: Revista Matematica de la Universidad Complutense deMadrid, Vol. 8, numero 1, 1995).

4. Arnold, V.I. (1994) On the topological properties of the Legendrian projections in the contact topologyof wavefronts, Sanel Petersburg Math J., vol. 6, n. 3, pp. 1-16.

5. Arnold, V.I. (1994) On the flattening point numbers of space curves. Preprint ISSN IML llO.1-467X,ISRN IML-R-I-94/95, Report Nl 1994/95, Institut Mittag-Leffler, 13 pp. (also in: Advances in Sov.Math., AMS, Providence, 1995, Sinai's volume).

6. Arnold, V.I. (1995) Geometry of spherical curves and algebra of quaternions, Russian Math. Surveys,50, n. 1, 3-68.

7. Arnold, V.I. (1995) Invariants and perestoikas of plane wavefronts, Trudy Steklov Math. Inst., 209,65 pp.

8. Arnold, V.I. (1995) Remarks on the extatic points of plane curves, Cahlers dc Mathcmatiques de ladeCIsion, n. 9529, CEREMADE, Uni',ersite Paris-Dauphine (URA CNRS 749), 26/06/95, pp. 1-16(also to appear in: Arnold's and Gelfand's seminars, Birkhauser, 1995).

HETEROCLINIC CHAINS OF SKEW PRODUCTHAMILTONIAN SYSTEMS

SERGEY V. BOLOTINDepartment of Mathematics and Mechanics, Moscow State UniversityMoscow 119899, Vorob 'evy Gory, Russia

1. Skew product Hamiltonian systems

Consider a time dependent Hamiltonian system

(1)

with configuration manifold M and phase space

P = T* M = {z = (q,p) I q E M, PET; M}

Here J : T; P --+ TzP is the operator corresponding to the symplectic structure dp 1\ dq.Fix a complete Riemannian metric II . lion M. Let H satisfy the following assumptions.

(HO) H E C~(P x IR) and H, \1zH, \1;H E CO(P x IR) are uniformly continuous onK x IR for any compact set K C P.

(HI) The Hessian \1~H of the Hamiltonian in momentum 11 E T; M is uniformlypositive definite on K x IR for any compact set K C P.

(H2) H is uniformly superlinear: H/llpll --+ 00 uniformly as Ilpll --+ 00.(H3) The phase flow 9T : P x IR --+ P x IR of system (1) is uniformly complete: for any

compact K C P and T > 0 there is a compact KT sllch that 9T(K x IR) c KT X ~ forITI

14

(F2) F is superlinear in momentum: F/llpil ---t 00 as IIpll ---t 00.(F3) The phase flow Gt : P x N ---t P x N is complete.(F4) F is quadratic in p for sufficiently large Ilpll.Condition (F4) can be omitted. Suppose that for some Xo E N, we have H(z, t) =

F(z, (/;t(xo)). Then H satisfies (HO)-(H4) and z(t) =

15

of the same dimension as M and there exist C, ,\ > 0 such that

The stable and unstable invariant sets of r equal

W(f) = {(z,x) E P x N I z E W(x)}.

Homoclinics to r lie in W+ (f) n W- (r) \ r. Hyperbolic invariant graphs are stable underperturbations of a skew product system.

Definition 2.2 An invariant graph r is called minimizing if it satisfies the Weierstrasscondition: there exists a function W E C'lvr (M x N) and a closed i-form ,\ on M such thatfor every x E N the function

V(q, x) = W'(q, x) -I- F(q, w(q, x), x), w(q,x) = ,\(q) -I- V"qW(q,x),

has a strict maximum h(x) at q = g(x) and f{x) = (g(x),w(g(x),x)).Here W' is the time derivative

W'(q, x) = lim(W(q, >,(x)) - W(q, x))/c.,-+0

(6)

We assume that the limit exists for all x E N and W' E Clt(M x N). For a smoothskew product system, W' = (V"xW,v). For example, an equilibrium r = {(qo,O)} x N is aminimizing graph if qo is a point of strict maximum of the function F(q, 0, x) for all x E N[17,11]. For a minimizing hyperbolic graph, the projections of Wl;c(x) to a neighborhoodof g(x) in Mare diffeomorphisms.

Let L E CfM(TM x N) be the Lagrangian

L(q,q,x) = max((p,q) - F(q,p,x)),p

p = V"qL(q,q,x) E T;M. (7)

We identify P x N with TM x N using the Legendre transform. If r is minimizing,performing the canonical transformation

P -t P - w, F -t F -I- W' - h, L -t L - ,\ - W -I- h,

we may assume that Fir = Llr = 0 and L > 0 everywhere on (P x N) \ r. From now onwe always make this assumption and also (FO)-(F3).

3. Existence of minimizing homoclinics

Let a-(t) = (z(t), >t{x)) be a homoclinic to a minimizing graph r. Denote I(t) = 7T'M{a(t))and IX(t) = 7T'M(ax(t)) = g(I/;t(x)). Then dist(l(t),'Yx(t)) -t 0 as t -t oo. Adding toII[a,b] - Ixl[a,b] short segments connecting I(b) with 'Yx(b) and Ix(a) with 'Y(a), we get aclosed curve in M. For a -t -00 and b -t 00, its homology class doesn't depend on a andb. It is called the homology class [a] E HI (M, Z) of the homoclinic trajectory a.

For a curve I: [a,b] -t M and x E N let

I~(r,x) =t L(r(t),i'(t),>t(x))dt

16

be the Hamilton action. For a homoclinic a, we put S(a) = I~c>.),,:I:). Note that I istranslation invariant: I~oo(TT,,1hX) == I~ooh,x), where TT,(t) = ,(t + T). In general,a < S(a) ~ 00. However, if r is hyperbolic, then S(a) < 00.Condition at Infinity. If M is noncompact, we always impose the following assumptionat infinity. Let I be the Maupertuis metric

I(q,q) = max{(p,q) I maxF(q,p,x) :S a}.p x

(8)

Then I(q, q) 2: a and I(q, q) lIlay vanish for q =I- a only if q E 7rJII (f). Thus I is a CO Finslermetric on M \ 7rM(f). From (7) and (8) it follows that L 2: I. Define t.he distance d(qo,qtlas the infimum of lengths of curves connert.ing qo with ql and vice versa:

We assume thal fOT' any qo E M and c > a the bo.ll Bc(qo) = {q E 1\1 I d(q,qo) ~ c} '/05compact. For example, t.his holds in the presence of st.rong force singularities [23].

Theorem 3.1 If f is a minimizing invariant graph, th.en: e:cist sequences of additive gen-emtors {ad of H j (1\1, 'iL) and act'ion minimizing homodinics {ad such that [0'1.'] = 0'1.'.

Rema1'/.;s. 1. In general t.he poillts :1'); = 7rN(aJ;(a)) are differcnt. Thus ak are doublyasymptotic to different trajectories a;r, in f.

2. In general not every homology class ill HdJl,1,'iL) contains homoclinics. For example,consider the system of n independent pendula. Then M = 1['n and the group HI (M, 'iL) ='iLn has 2n additive generators a1, ... ,an . Let r be the upper equilibrium. Then everyhomoclinic to f helongs to one of 3n - 1 homology classes L:i'=l [iai with [i = or l.

3. If t.he image of the group 7rdN) in 7r1 (1\1) under 9 is contained in the center of7rj (M) (for examplc, f is an equilibrium), we can define the homotopy class [a] E 7r1 (1\1)of a hornoclinic a. Then Theorem 3.1 holds for homot.opy classes in 7r1(M) if we replaceadditive generators by multiplicat.ive ones. See [7] for periodic and [3, 4] for quasiperiodicand almost periodic cases.

PI'oof of TheoTe'lIl 3.1. We follow [7, :3]. Let AC([O, 1], M) be t.he set of ahsolutely continuouscurves w : [0,1] ---+ M with the standard topology and

n = {a = (w, :E, Tl E AC([O, 1], 1\1) x N X rn;,+ I w(O) = g(:r), w(1) = g(T(X))}, (9)

Let S : n ---+ rn;, U { +oo} be the action functional

S(a) = S(w,x,T) = I6h,x), ,(t) = w(tjT).

Since r is minimizing, S 2: 0, and if S(a) = 0, then I'(t) == I'x(t.). Define the homologyclass [a] E A = H)(M,'iL) of 0'= (w,:r,T) Elias the class of the cycle w - I'xl[o,n Weobtain a function R on A:

R(a) = inf S(a),n,., llo = {a E n I [a] = a}.

Lemma 3.1 We have R(O) = 0 and R(a) > for a =I- O. For' any c > 0, the set {a E A IR(a) :S c} is finite.

17

Indeed, the condition at infinity and inequality L ~ 1 imply that if S(a) ~ c, then w iscontained in the compact set Kc = UxENBc(g(x)).

Lemma 3.2 There exists a sequence of additive generators {ak} of A such that if ak ={3 + {3' for some {3, {3' E A, then R({3) ~ R(ak) or R({3') ~ R(ak)'P1'00f. By Lemma 3.1, we can define ak E A by induction setting

(10)

where [al,' .. , ak-d is the semigroup generated by aj, ... ,ak-j If ak = {3 + {3', we haveeither {3 fi- [al' ... , ak- tl or {3' fi- [a], ... , ak- d Thus the statement of Lemma 3.2 followsfrom (10).

Lemma 3.3 For all k thel'e is a homoclinic ak with [akl = ak and S(akl = R(ad.

Proof. Denote a = ak and let c > R(a). Since L is superlinear and a i- 0, we haveS(a) -+ 00 as T -+ 0 for a = (w, x, T) E no' The set

0 the functional Se = S + cT has a minimum a == (w, x, T) on no andminna Se -+ R(a) as c -+ O. Let ,(t) = w(tjT) and p(t) = V'qL(r(t),-r(/.),r(x)). Then(r(t),p(t), >t{:z;)) is a trajectory of system (2).

Denote w(x) = V'pF(J(x),x). Then -rAt) = W(1(1:)). Using the first variation formula[2] for V'TS(W, x, T) = V'TS(W, T(~), T) = c, where x = T(~), we obtain

(p(T),w(tjrr(x))) - F(g(T(x)),p(T),tjrr(x)) = (p(O),w(x)) - F(g(x),p(O),x) = c. (11)

Since r is minimizing, for any x E N the convex function L(g(x), ., x) has a minimum 0at w(x). From (7) and (11) it follows that

II'Y(O) - w(x)11 + II'Y(T) - W(T(1;))11 ~ C..;E., c > O. (12)Lemma 3.4 There exist small p for all t E [a, b]. Hence L ~ const > 0 for t E [a, b], and so b - ais bounded. By (12), a -+ 00 and T - b -+ 00 as c -+ O.

18

By the condition at infinity, , is contained in a compact set K c C M. Since N iscompact, there exists a sequence I;; -+ 0 such that x -+ Xo E N, b - a -+ T > 0 and thecurve T_ a, is uniformly convergent to a curve ,0 :IR -+ M on any finite interval. Hencedist (ro(0),9(Xo)) = dist (rO(T), 9(

19

is a convergent integral and for any J1, > 0 and sufficiently large T > 0,

I jT+b-a I

1- -T h(ao(t)) dt < J1,.

The trajectory T_aa converges to 0'0 uniformly on any finite interval. Since b-a is bounded,for given T > 0 and sufficiently small E: > 0, we have

Lemma 4.1 There exist constants CI,2 > 0 such that

Ifo a-T

h(a(t)) dtl < v, IrT h(a(t)) dtl < V,

JIJ+T

Equation (15) yields III < J1, + 2v. Since T > 0 is arbitrary and E: -t 0, equality (14) isproven. _

Proof of Lemma 4.1. The functions r(z, x) = dist (z, Wl~c(x)) are well defined in a smallneighborhood of r in P x N. Since hlr = 0, we have h = O(r + + 1_). Hyperbolicity of rimplies that the functions 1(t) = r(a(t)) satisfy the inequalities

o :'S t :'S a.

We obtain

We used here that r +(a) is bounded as E: -t 0 and r _(0) is of order ft by (12). _

Since the homoclinics ak in Theorem 3.1 were minimizing, a manifold structure onfl wasn't needed. Now let M be simply connected and r a C l minimizing hyperbolicinvariant graph of a smooth skew product system (3). Then it. is convenient to define thefunction space fl in (9) as a submanifold of the Hilbert manifold W I ,2([0, 1]' M) x N X lR.+.Under condition (F4), S E C 1(fl), but doesn't satisfy the PS condition. Hyperbolicity of rimplies that the set of minimum points flo ~ N X lR.+ is a nondegenerate critical manifoldof zero index. Obviously, S has no other critical points in fl.

Let M be noncontractible, for example, compact. Then 7r;(M) of 0 for some i > 1.Hence 7ri-1 (fl, flO) of 0 and so S has nonzero minimax levels.

Theorem 4.2 For any minimax level c of S on fl, there exists a chain of homoclinics0'1, ... ,at to r such that

2:S(ak) = c,k

2:t:.uJ = o.k

(16)

(17)

20

In general ak are doubly asymptotic to different trajectories in r.Sketch of the pmof. First we find a critical point a f on a level Sf = Cf of the functionalSf = S + ET for small E > O. The proof of the existence is a modification of the proofof Theorem 3.1. Hyperbolicity of r is used for constructing a pseudogradient flow of thefunctional Sf near ~o, where the completeness and PS condition fail. The technical detailsare similar to that of [10] for homoclinics to invariant tori of Hamiltonian systems and [3]for quasiperiodic systems.

In contrast to Theorem 3.1, in general in the limit E -+ 0 we don't obtain a homoclinicof action c but a chain of hOllloclinics satisfying (16)-(17). The proof of (16) follows theusual concentration compactness procedure [13,15]. Equality (17) follows from (15) as inthe proof of Theorem 3.1.

Remark. An alternative approach is to use the methods developed in [13, )(J, 24, 25]and other papers for homoclinics to equilibria of time periodic systems. Embed M in aEuclidean space Iftd. Then we can subtract points of M. Let

(18)

If r is reduced to an equilibrium, A ~ ~(M) x N, where ~(M) is the Hilbert manifoldof loops [16]. If (F4) is satisfied, the action functional S(w) = I""'oo(r,x) is finite andtranslation invariant: 80TT = 8, where TT(r,.r,) = (TTl'

21

6. Multibump homoclinics

In this section we assume that r is hyperbolic and the flow

22

Theorem 6.2 If the set KnIT is disconnected, then assertions of Theorem 6.1 hold.

Proof. The projections of the local stable and unstable sets W/;c(f) are homeomorphismsto a small tubular neighborhood U of L: = JrMxN(f) in M x N. Let

W/;c(f) = {(q,p,x) I (q,x) E U, p=p(q,x)}.

Let a(t) be trajectories with initial points a(O) = (q,p(q,x),x) E WI~c(r). Put

The functions S : U -7 IR satisfy the Hamilton-Jacobi equations

where S~ is the time derivative (6). The calculus of variations implies that

S+(q,x) = inf{IO'(r,x) 1,(0) = q}, S_(q,x) = inf{I~cx,(r,x) 1,(0) = q}. (19)

Let cP : IT -7 IT and (j> : P x IT -7 P X IT be the Poincare maps of the flow cPt and system(2) respectively. Then cPk(x) = cPrA-{x)(x) and (j>k(z,x) = GrA-{x)(z,x), where Tdx) is thek-th return time to IT. Put V = un (M x IT) and

k E Z.

Let U be sufficiently small and I E Z+ :;ufficiently large.

Lemma 6.1 For all n 2 I ther'e exist continuous functions y = y(q_,x_,q+,x+) ET;M on Zn such that (j>n(q_,y_,x_) = (q+,y+,x+). The action of the tmjector'y a(t) =Gt (q_, y_, x_), a ::::: t ::::: Tn (1;_)' equals

I~"(x-)(a) = gn(tf-,L,Cf ,x+) = S_(Cf_,:L) + S+(q+,x+) + un(q_,x_,q+,x+), (20)

where Ilunllco ::::: J.L1 -7 a as I -7 00.This follow:; from the fact that the projections of Wl~c(x) to Mare diffeomorphisms to

a neighborhood of g(x). Indeed, hyperbolicity of r implies that each point in P x N closeto f tends exponentially to W- (r) as t -7 00 and to W+ (r) as t -7 -00.

Let K 0 C P x IT be a connected component of the set KnIT not containing r n (P x IT).There exists m > a such that D = JrMxn(j>m(Ko)) C V. Let V C V be closedconnected neighborhoods of the sets D not intersecting L:. Define a map 'l/J : V_ -7 V-tby 'l/J(Cf-,X-) = JrMxn(j>k(Cf_,P_(q_,x_),x_)), where k = 2m. If (Cf-,L) E D_, then'l/J(q-,x_) E D+ and a(t) = Gt(q-,p-,x_) is a homoclinic trajectory of the homologyclass a.

Let

where the homology class [r] E HdM,Z) of the curve, : [O,Tdx)] -7 M is definedsimilarly to the homology class of a homoclinic orbit. Consider the function

f(Cf-,L,q+,x-t) = S_(Cf-,L) + hdq_,L,q+,x+) + S-t(q+,x+) - R(a) (22)

OIl the set Y = Zk n (V- x V-t). Equations (19) and (21) yield

23

Lemma 6.2 Let the neighborhoods V of D be small enough. Then f has a minimum 0on Y and !(q_,x-,q+,x+) = 0 iff (q,x) E D and 'I/J(q-,x_) = (q+,x+).

Take sufficiently large l > 0 and integers nl, ... , ns-I 2: l. Let

The points in X C y s satisfy the condition Xt E A = 1rn(V-) and xi = m;(xt )E A fori = 1, ... , s, where m; = L;~~ nj' Define a function :F on X by the formula

s s-1

:F = S_(qt,xt ) + Lhk(q;-,xi,qt,xn + L9n i(qt,xt,qi+l>xi+l) + S+(x;-,q;).;=1 ;=1

From Lemma 6.1 it follows that if l is large enough, :F is well defined and continuous. By(20) and (22),

s s-I

:F = L j; + L uni (qt, xt, qi+l> xi+I)';=1 ;=1

Lemma 6.3 There exist infinitely many nl, ... , ns-I 2: l such that :F has a minimumpoint in the interior 0/ X.Proof. Let 6 > 0 be the infimum of / on aY and E E (0,6). By Lemma 6.2, there existsa neighborhood W C V_ of D_ such that /(q-,x_,'I/J(q_,x_)) < E for (q_,x-) E W.Let B = 1rn(W) and C = 1rn(D_). Denote d = dist (C, aB)/:3 and B l = {x E N Idist(x, C) < d}, B2 = {x E N I dist (x, C) < 2d}. Let 60 > 0 be the infimum of! on theset of points in Y with x_ rt B 1. We take EO E (0,60 ) and a neighborhood Wo of D_ suchthat /(q-,x_,'I/J(q-,x-)) < EO for (q_,x_) E Woo Put Bo = 1rn(Wo)

Since the flow t is almost periodic, for any v > 0 there exist n 1, ... , n s-l 2: l suchthat dist(mi(x),x) < v for all x E A and i = 1, ... ,s. Let v < min{d,dist(C,aBo)}.Then for i = 1, ... , s, we have

In particular, X is nonempty. Since X is compact, :F has a minimum. We must showthat the minimum point lies in the interior of X. First we prove that xi E B for all i.By (24), if xi rt B for some i, then Xt rt B2 . Hence xj rt B I for all j and so Ii 2: 60On the other hand, taking Yl E C, we obtain Yj = ;ffij (Yt) E Bo for j = 1, ... , n.Hence we can choose rj E M so that (rj,yjl E Woo Let (rj,yj) = 'I/J(rj,yjl. Then/(rj,Yj,rj,yj) < EO. By (23), for large I we get a point in X with smaller value of:F.

Thus xi E B for i = 1, ... , s. Suppose that (q;-, xi) E av_ for some i. Then j; 2: 6.Take r;- EM so that (ri,xi) E W, put (rt,xn = 1/J(ri,xi) and replace q; by r; in(23). Then we decrease j; by at least 6 - E > 0 not changing other terms /j' Since Uj arearbitrary small, we obtain a point in X with smaller value of :F.

The minimum point of :F in the interior of X yields a mult.ibump homoclinic. Thefirst part of Theorem 6.2 is proven. For a C 1 invariant graph r and a C I stable flow t,we obtain a critical point (--y,x) of the functional S on A/IR, where x = xl' Let 11 be

24

a symmetry field. Differentiating S in the direction of the vector u(x) E TxN, we get!:!.uJu = (\7 xS,u(x)) = O.

7. A version of diffusion

Consider a smooth skew product system (3) with Hamiltonian (4). To prove the existenceof trajectories with a given change of J : P -+ C* is a simple version of the problem ofArnold diffusion [1, 12]. We study jumps !:!.uJ along homoclinics a to the minimizing graphr. If !:!.uJ f= 0, from the point of view of the Hamiltonian system on P, the trajectory ais a hetemclinic orbit (the limit sets n (a) C P are different).

Let all assumptions of Theorem 6.2 be satisfied arid Ko be a connected component ofthe set KnIT in Theorem 6.2. Denote ITo = 7r1l(Ko) and let E C Hl(N,~) be the image ofthe Cech homology group of ITo under the inclusion to N. Let A be any

25

Acknowledgements

The author is grateful to the Russian Foundation of Basic Research and INTAS for financialsupport.

References

1. Arnold, V.I. (1964) Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk.SSSR, 5, 581-585

2. Arnold, V.I., Kozlov, V.V and Neishtadt, A.I. (1988) Mathematical Aspects of Classical and CelestialMechanics. Dynamical Systems, Vol. III, Springer-Verlag, New York-Heidelberg-Berlin

3. Bertotti, M.L. and Bolotin, S.V. (1995) Homoclinic solutions of quasiperiodic Hamiltonian systems,DiffeTential and Integral equations, 8, 1173-1760

4. Bertotti, M.L. and Bolotin, S.V. (1995) A variational approach to homoclinics of almost periodicHamiltonian systems, to appear in Communications in Applied NonlineM Analysis

5. Bessi, U. (1994) An approach to Arnold's diffusion through the calculus of variations, PTepnnt6. Bolotin, S.V. (1978) Libration motions of natural dynamical systems, Vestnik Moskov. Univ. SeT.

Matern. Mekh., 6, 72-777. Bolotin, S.V. (1983) The existence of homoclinic motions, Vestnik M9skuv. Uni". SeT. I Matern.

Mekh., 6, 98-1038. Bolotin, S.V. (1990) Homoclinic orbits for invariant tori in the perturbation theory of Hamiltonian

systems, PTikl. Matern. i Mekhan., 54, 497-5019. Bolotin, S.V. (1992) Homoclinic trajectories of minimal tori of Lagrangian syst.ems, Vestmk Muskov.

Uni". SeT. I Matern. Mekh., 6, 34-41to. Bolotin, S.V. (1992) Homoclinic orbits t.o invariant tori of Hamiltonian ';ystems, P,-epnnt (to appear

in AMS Advances in Soviet. Mathematics)11. Bolotin, S.V. and Kozlov, V.V. (1980) On asymptotic solutions of t.he equations of dynamics, Vestmk

Moskov. Univ. SeT. I Matern. Mekh., 6, 72-7712. Cierchia, L. and Gallavotti, G. (1994) Drift and diffusion in phase space, Ann. Inst. Hem'i PoincaTe,

60, 1-14413. Coti Zelati, V., Ekeland, l. and Sere, E. (1990) A variat.ional approach to homodinic orbit.s in Hamil-

tonian syst.ems, Math. Ann., 288,133-16014. Calderoli, P., Coti Zelati, V. and Nolasco, M. (1995) Multibump solutions for a class of second order

almost periodic Hamiltonian systems, P"epnnt15. Coti Zelati V. and Rabinowitz, P.H. (1991) Homoclinic orbits for second order Hamiltonian systems

possessing superquadratic potentials, .I. AmeT. Math. Soc., 4, 693-72716. Giannoni, F. and Rabinowitz, P.H (1993) On the multiplicity of homoclinic orhits on Riemannian

manifolds for a class of second order Hamiltonian systems, NoDEA, 1, 1. 4917. Hagedorn, P. (1975) Uber die Instabilitiit konservativer Systeme mit Gyroskopischen Kraften, ATch.

Rat. Mech. Anal., 58, 1-918. Kozlov, V.V. (1985) Calculus of variations in large and classical mechanics, Uspekhl Matern. Nauk,

40, 33-6019. Mather, J. (1991) Action minimizing invariant measures for positive definite Lagrangian systems,

Math. Z, 227, 169-20720. Mather, J. (1993) Variational construct.ion of connecting orbits, Ann. Inst. FouTier 43, 1349-138621. Meyer, K.R. and Sell, G.R. (1989) tVIelnikov transforms, Bernoulli bundles and almost periodic per-

turbat.ions, Trans. of the Arne,.. Math. Soc., 314, 63-10522. Nemytsky, V.V. and Stepanov, V. V. (1960) Qualitative theoTy of diffeTential equations, Princeton

University Press, Princeton23. Rabinowitz, P.H. (1994) Heteroclinics for reversible Hamilt.onian systems, E,gudic TheoTy and Dy-

namical Systems, 14, 817-82924. Sere, E. (1992) Existence of infinitely many homoclinics ill Hamilt.onian systems, Math. Z., 209, 27--4225. Sere, E. (1993) Looking for t.he Bernoulli shift., Ann. Inst. H. Poincani, Anal. Non Lineaire, 10,

561-59026. Serra, E., Tarallo, M. and Terracini, S. (1994) On t.he existence of homoclinic solutions for almost

periodic second order systems, PrepTint27. Sim6, C. (1994) Avcragillg under fast quasiperiodic forcillg, Hamiltonian mecha7llcs: Integrability and

Chaotic behavior, Plenum Press, 13-34

ORDER AND CHAOS IN 3-D SYSTEMS

G. CONTOPOULOS, N. VOGLIS AND C. EFTHYMIOPOULOSDepartment of Astronomy, University of AthensPanepistimiopolis, GR -15784 Athens, Greece

1. Introduction

The study of order and chaos in 3-D systems is a subject of great current interest. Al-though many aspects of these problems in 2-D systems are more or less understood, thecorresponding problems in systems of 3 degrees of freedom are still largely unexplored.

In the present lectures we will cover briefly the following topics.(1) 3-D Stackel potentials (integrable).(2) A new transition from order to chaos in 3-D systems. Complex Instability.(3) Order in the distribution of 3-D periodic orbits.(4) Asymptotic curves of 3-D unstable orbits.(5) Spectra of short-time Lyapunov numbers.(6) Chaos in a particular 3-D system. The Mixmaster UniversE' model.

2. 3-D Stackel potentials

An important class of integrable 3-D systems are the Stackel potcntials (Stackel 1890,1893), which are of the form (Weinacht 1924)

v = F--,l:....:.(--,A)--'--_(A - fl)(A - v) (fl - v)(1-l - A) (v - A)(V - I-l)"

(1)

(2)

The variables A,fl,1/ are ellipsoidal coordinates, which are related to the cartesian coordi-nates X,Y,Z through the equation

x 2 y2 z2--+--+--=1U - 0,2 'Il - b2 '11, - c2 .

The 3 roots u of (2) are A,I-l,1/ where A ~ 0,2 ~ fl ~ b2 ~ v ~ c2 with a,b,c constants.The Stackel potentials havc many important applications in galactic dynamics, because

they can represent plausible models of galaxies with convenient choices of the functionsF 1(A), F2(fl) , F,3(1/). In particular one can construct self-consistent models of galaxies(de Zeeuw 1985; de Zeeuw et al. 1987, Hunter 1988).

The main types of orbits in such potentials are shown in Fig. 1. They are (a) boxorbits, (b) short-axis tube orbits, (c) inner long-axis tube orbits and (d) outer long axistube orbits.

In a recent paper (Contopoulos 1994) we have shown that there are 32 different typesof orbits in any model (1), with continuous functions F 1(A), F2 (I-l) , F3 (v).

26

27

Figure 1. The regions filled by the main 4 typesof orbits in a Stackel potential (after Statler 1987).

Du

5

.o,!--~-'J--c

Figure 2. Regions of the parameter space (c,1])where the central periodic orbit (1a) is stable (5),simply unstable (U), doubly unstable (DU), or com-plex unstable (t:.)(.-.-) boundaries between the var-ious regions. Along th.~ continuous lines 1/2, 1/3,1/4 we have bifurcations of families of periodic or-bits of periods 2,3,4. These lines are tangent to theboundary of t:..

The main drawback of the Stackel models is that they represent, in general, nonro-tating models. Thus they are only appropriate as approximate models of slowly rotatinggalaxies, like elliptical galaxies. The only nontrivial rotating Stackel potential was studiedby Contopoulos and Vandervoort (1992), but this potential cannot. be generalized to moregeneral galactic models.

Furthermore fast rotating systems, like barred galaxies, seem to have an importantfraction of chaotic orbits, which are, of course, absent from the SUickel potentials.

3. Transition to Chaos in 3-D Systems. Complex Instability

Generic 3-D systems can be generated from 2-D systems by changing a parameter. If theoriginal 2-D system is nonintegrable, it contains unstable periodic orbits that generatechaos through their homoclinic and heteroclinic tangles. The main mechanism for gen-erating a large degree of chaos in 2-D systems is by the interaction of infinite unstableperiodic orbits, generated by period doubling bifurcations (Feigenbaum 1978, Coullet andTresser 1978). If the 3-D system is generated from a largely chaotic 2-D system it has alsolarge chaotic regions.

However in 3-D systems we may have a new type of transition from stability to in-stability. This is complex instability. Namely the eigenvalues of a stable orbit may collidein pairs on the unit circle, outside the real axis, and from then on (as the perturbationincreases) they may move inwards and outwards from the unit circle, giving 4 eigenvaluesthat are complex conjugate in pairs, and also inverse in pairs

'\1,2 = p(cos 0 i sin 0) , ,\34= ~(cos (I i sinO)., p (3)

Whether or not collision of eigenvalues leads to complex instability is dealt with bythe Krein (1980) - Moser (1988) theorem.

When a transition to complex instability occurs, in general we do not have a bifurcationof other families of periodic orbits. Thus the introduction of instability and the resulting

28

Figure 3. The projections on the XI, 12 plane ofsuccessive Poincare consequents of an orbit closeto a complex unstable periodic orbit in the Hamil-tonian (4) with A = 0.9, B = 0.4, C = 0.225,E: = 0.56, 'I = 0.2 and H = 0.0765. (0) numericalpositions, (+) theoretical positions.

'00 f--=:;~~i~-Ix

-2 00h~~;::S;~~~~-2.50 -050 O. 0

x

FiyUl'e 1,. Continuation of the (asymptotic) curveof Fifl. 3. This curve reaches a maximum distancefrom the periodic orbit and then turns inwards.

chaos does not follow a cascade of infinite period-doubling bifurcations. Only in exceptionalcases, and along particular paths in the parameter space we may have such period doublingsequences. We have studied, in particular, the Hamiltonian

(4)

and we have found period doubling mainly along the axes T/ = 0, and E = 0, which repre;;ent2-D systems. The central periodic orbit (called 1a; Contopoulos 1994) generates abo doubleperiodic orbits along two lines that are tangent to the boundaries of the complex unstableregion Ll (the region of the parameter space (E,T/) in which the central periodic orbit iscomplex unstable, Fig. 2). We have also the generation of period-3, period-4 etc. orbitsalong the lines 1/3 and 1/4, and so on. Some of these lines arc tangent to Ll, but we stressthat the bifurcating families are not generated only at the tangent points with Ll. Wenotice, further, that there are no bifurcating families along the boundary of the complexnnstable region Ll with the doubly unstable region DU. Finally, inside the complex unstableregion Ll there arc dashed lines with rational ratio :/rr = 1/2, or 1/3, or 1/4 etc. Alongthese lines the orbits described 2, or 3, or 4 etc. times arc douhly unstable. But theseorbits do not gener