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Singularity theory for non-twist KAM tori PDF

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EMOIRS M of the American Mathematical Society Volume 227 • Number 1067 (third of 4 numbers) • January 2014 Singularity Theory for Non-Twist KAM Tori A. Gonza´ lez-Enr´ıquez A. Haro R. de la Llave ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society EMOIRS M of the American Mathematical Society Volume 227 • Number 1067 (third of 4 numbers) • January 2014 Singularity Theory for Non-Twist KAM Tori A. Gonza´ lez-Enr´ıquez A. Haro R. de la Llave ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society Providence, Rhode Island Library of Congress Cataloging-in-Publication Data Gonz´alez-Enr´ıquez, A. (Alejandra), 1967- author. Singularity theory for non-twist KAM tori / A. Gonz´alez-Enr´ıquez, A. Haro, R. de la Llave. pages cm – (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; number 1067) “January 2014, volume 227, number 1067 (third of 4 numbers).” Includes bibliographical references. ISBN 978-0-8218-9018-9 (alk. paper) 1. Bifurcation theory. 2. Perturbation (Mathematics) 3. Ergodic theory. I. Haro, A. (Alex), 1969- author. II. de la Llave, Rafael, 1957- author. III. Title. QA380.G66 2014 ′ 515 .39–dc23 2013035976 DOI: http://dx.doi.org/10.1090/memo/1067 Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. Beginning with the January 2010 issue, Memoirs is accessible from www.ams.org/journals. The 2014 subscription begins with volume 227 and consists of six mailings, each containing one or more numbers. Subscription prices are as follows: for paper deliv- ery, US$827 list, US$661.60 institutional member; for electronic delivery, US$728 list, US$582.40 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. If ordering the paper version, add US$10 for delivery within the United States; US$69 for outside the United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number. Back number information. For back issues see www.ams.org/bookstore. Subscriptions and orders should be addressed to the American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by payment. Other correspondence should be addressed to 201 Charles Street, Providence, RI 02904-2294 USA. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to Contents Part 1. Introduction and preliminaries 1 Chapter 1. Introduction 3 1.1. Towards a singularity theory for KAM tori 3 1.2. Methodology (a brief description) 5 1.3. Outline of this monograph 8 Chapter 2. Preliminaries 11 2.1. Elementary notations 11 2.2. Geometric preliminaries 12 2.3. Symplectic deformations and moment maps 16 2.4. Analytic preliminaries 17 2.5. Cohomology equations 19 Part 2. Geometrical properties of KAM invariant tori 21 Chapter 3. Geometric properties of an invariant torus 23 3.1. Automatic reducibility 23 3.2. Geometric definition of non-twist tori 25 3.3. Intrinsic character of the reducibility and of the torsion 26 Chapter 4. Geometric properties of fibered Lagrangian deformations 29 4.1. The potential of a fibered Lagrangian deformation 29 4.2. A parametric version of the potential 34 Part 3. KAM results 37 Chapter 5. Nondegeneracy on a KAM procedure with fixed frequency 39 5.1. Approximate reducibility of approximately invariant tori 39 5.2. Dummy and modifying parameters 41 Chapter 6. A KAM theorem for symplectic deformations 51 6.1. Functional equations and nondegeneracy condition 51 6.2. Statement of the KAM theorem 52 6.3. Proof of the KAM Theorem 53 Chapter 7. A Transformed Tori Theorem 67 7.1. Nondegeneracy condition 67 7.2. Statement of the Transformed Tori Theorem 68 7.3. Proof of the Transformed Tori Theorem 70 iii iv CONTENTS Part 4. Singularity theory for KAM tori 73 Chapter 8. Bifurcation theory for KAM tori 75 8.1. Classification of KAM invariant tori 75 8.2. Local equivalence of Bifurcations diagrams 77 Chapter 9. The close-to-integrable case 81 9.1. The integrable case 81 9.2. Persistence of invariant tori in quasi-integrable systems 82 9.3. Unfolding non-twist tori 86 9.4. The Birkhoff potential and the potential of an invariant torus 86 Appendices 91 Appendix A. Hamiltonian vector fields 93 A.1. Cohomology equations 93 A.2. Automatic reducibility of invariant tori 94 A.3. Families of Hamiltonians and moment maps 97 A.4. Potential and moment of an invariant FLD 98 A.5. Transformed Tori Theorem 100 A.6. A KAM Theorem for families of Hamiltonians 102 Appendix B. Elements of singularity theory 105 Bibliography 111 Abstract In this monograph we introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency in parameter-dependent Hamiltonian sys- tems. It is based on Singularity Theory of critical points of a real-valued function which we call the potential. The potential is constructed in such a way that: nonde- generate critical points of the potential correspond to twist invariant tori (i.e. with nondegenerate torsion) and degenerate critical points of the potential correspond to non-twist invariant tori. Hence, bifurcating points correspond to non-twist tori. Invariant tori are classified using the classification of critical points of the po- tential as provided by Singularity Theory: the degeneracy class of an invariant torus is defined to be the degeneracy class of the corresponding critical point of the potential. Under rather general conditions this classification is robust: given a family of Hamiltonian systems (unperturbed family) for which there is a Hamil- tonian with an invariant torus (unperturbed torus) satisfying general conditions, explicitly given in the monograph, we show that for any sufficiently close family of Hamiltonian systems (perturbed family) there is a torus (perturbed torus) that is invariant for a Hamiltonian of the perturbed family and such that both per- turbed and unperturbed tori have the same frequency and belong to the same degeneracy class. Our construction is developed for general Hamiltonian systems and general ex- act symplectic forms. It is applicable to the case in which a bifurcation of invariant Received by the editor May 24, 2011, and, in revised form, March 6, 2012. Article electronically published on June 24, 2013. DOI: http://dx.doi.org/10.1090/memo/1067 2010 Mathematics Subject Classification. Primary 37J20,37J40. Key words and phrases. Parametric KAM theory, non-twist tori, quasi-periodic bifurcations, Lagrangian deformations, moment map, Singularity theory for critical points of functions. A. G.-E. has been funded by a B. de Pin´os Fellowship, and her visits to the Department of Mathematics of the University of Texas at Austin have been funded by NFS grants. A. H. has been funded by the Spanish grants MTM2006-11265 and MTM2009-09723 and the Catalan grants 2005-SGR-01028 and 2009-SGR-67. R. L. has been funded by NHARP0203 and NFS grants. His visits to Barcelona have been funded by MTM2009-06973. A. G.-E. and A. H. thank the hospitality of the Department of Mathematics of the University of Texas at Austin. Thanks also to Michael Maudsley for his English grammar revision of the manuscript. Affiliations at time of publication: A. Gonz´alez-Enr´ıquez, Dept. de Matem`atica Apli- cada i An`alisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain, email: gon- vi ABSTRACT tori has been detected (e.g. numerically) but the system is not necessarily written as a perturbation of an integrable one. In the case that the system is written as a close-to-integrable one, our method applies to any finitely determinate singularity of the frequency map for the integrable system. Part 1 Introduction and preliminaries CHAPTER 1 Introduction 1.1. Towards a singularity theory for KAM tori KAM Theory was originally concerned with the problem of the persistence of quasi-periodic solutions under perturbations of integrable systems (i.e. systems for which for every action there is a quasi-periodic solution). In the pioneering works [4, 55,64] the persistence of quasi-periodic motions is guaranteed, provided the integrable system satisfies a nondegeneracy condition which is known as the twist condition [64] or Kolmogorov condition [4, 55]. The twist condition requires the frequency map to be a local diffeomorphism. We recall that the frequency map of an integrable Hamiltonian system takes actions into frequencies (see e.g. Appen- dix 8 in [7]). There has been considerable progress in extending the theory to a vari- ety of situations [22,31]. Since the fundamental work [68], significant advances have been made in weakening the nondegeneracy conditions [26,29,44–46,73,75,77,93] and in understanding the role of extra parameters [18,19,80]. A very efficient strat- egy to deal with degenerate problems (i.e. problems for which the twist condition is violated) has been to add extra parameters and extra conditions. In this monograph we give sufficient conditions under which a degenerate quasi- periodic solution persists. A novelty is that the persistent motion has the same frequency and belongs to the same degeneracy class as the unperturbed one. This is somewhat different from previous results on the persistence of KAM tori in degenerate systems [26, 29, 44–46, 73, 75, 77, 93], where neither the frequencies nor the class of degeneracy of the persistent tori are controlled. One motivation for our study is the widely held belief that in certain physical systems degenerate quasi-periodic motions with prescribed frequency are more “ef- fective barriers for transport” than nondegenerate ones [3, 16, 32, 33, 36, 37, 76]. Considering extra parameters is very natural in physical problems for which quasi- periodic solutions are desirable. For example, in the design of plasma confinement devices, the designers of the machine may make quantitative choices regarding the size of pieces, currents, etc. to ensure that the desired quasi-periodic solutions are present. Hence, it has become a design goal to adjust design parameters so that the system presents degenerate quasi-periodic motions. A main goal of this monograph is to provide an effective tool to be used in such design goal. The precise definition of “effective barriers for transport” is beyond the scope of this manuscript. Our methodology uses techniques from Symplectic Geometry [24, 63, 87–89] and from KAM theory. We develop a KAM theory for symplectic deformations, which may be of independent interest. We prove several geometrical properties of invariant tori for symplectic deformations with prescribed frequency (i.e. tori supporting quasi-periodic solutions with frequency previously chosen). Persistence 3

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