Singularity theory for non-twist KAM tori

128 Pages·2014·0.8 MB·English

by  Gonzalez-Enriquez A., Haro A., de la Llave R.

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
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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