Dynamics Reported: Expositions in Dynamical Systems

242 Pages·1993·7.376 MB·English

by  Chr. Genecand (auth.), C. K. R. T. Jones, U. Kirchgraber, H. O. Walther (eds.)

Dynamics Reported
Expositions in Dynamical Systems
Board of Editors
H. Amann (Zurich), P. Brunovsky (Bratislava), S. N. Chow (Atlanta), C. K R.T. Jones
(Providence), U. Kirchgraber (Zurich), H. Matano (Tokyo), J. Mawhin (Louvain-la
Neuve), R.D. Nussbaum (NewBrunswick), C. Robinson (Evanston), H.O. Walther
(Munchen), E. Zehnder (Zurich)
Advisory Board
L. Arnold, H. Broer, A. Coppel, B. Fiedler, J. Grasman, E. Hairer, J. Henrard, R.A.
Johnson, H. Kielhofer, A. Lasota, M. Misiurewicz, R. Moeckel, R.E. O'Malley, KJ.
Palmer, H.O. Peitgen, F. Przytycki, K Rybakowski, D.G. Saari, J.A. Sanders, A.N.
Sharkovsky, J. Scheurle, K Schmitt, A. Vanderbauwhede, J. Waldvogel, J. A. Yorke
Dynamical Systems are a rapidly developing field with a strong impact on applica
tions. Dynamics Reported is a series of books of a new type. Its principal goal is to
make available current topics, new ideas and techniques. Each volume contains about
four or five articles of up to 60 pages. Great emphasis is put on an excellent pre
sentation, well suited for advanced courses, seminars etc. such that the material
becomes accessible to beginning graduate students. To explain the core of a new
method contributions will treat examples rather than general theories, they will de
scribe typical results rather than the most sophisticated ones. Theorems are accompa
nied by carefully written proofs. The presentation is as self-contained as possible.
Authors will receive 5 copies of the volume containing their contributions. These
will be split among multiple authors.
Authors are encouraged to prepare their manuscripts in Plain TEX or LA TEX.
Detailed information and macro packages are available via the Managing Editors.
Manuscripts and correspondence should be addressed to the Managing Editors:
C. K. R. T. Jones H. O. Walther
Division of Applied Mathematics Mathematics
Brown University Ludwig-Maximilians University
Providence, Rhode Island 02912 W -8000 Munich
USA Federal Republic of Germany
e-Mail: [email protected] e-Mail: Hans-Otto. Walther
@mathematik.
U. Kirchgraber
uni-muenchen.dbp.de
Mathematics
Swiss Federal Institute
of Technology (ETH)
CH-8092 Zurich, Switzerland
e-Mail: [email protected]
C.K.R.T. Jones U. Kirchgraber H.O. Walther
(Managing Editors)
Dynamics Reported
Expositions in Dynamical Systems
New Series: Volume 2
With Contributions of
H. S. Dumas Chr. Genecand 1. Henrard 1. Komornfk
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
ISBN-13: 978-3-642-64755-0 e-ISBN-13: 978-3-642-61232-9
DOl: 10.1007/978-3-642-61232-9
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Preface
DYNAMICS REPORTED reports on recent developments in dynamical systems.
Dynamical systems of course originated from ordinary differential equations.
Today, dynamical systems cover a much larger area, including dynamical pro
cesses described by functional and integral equations, by partial and stochastic
differential equations, etc. Dynamical systems have involved remarkably in recent
years. A wealth of new phenomena, new ideas and new techniques are proving
to be of considerable interest to scientists in rather different fields. It is not
surprising that thousands of publications on the theory itself and on its various
applications are appearing.
DYNAMICS REPORTED presents carefully written articles on major sub
jects in dynamical systems and their applications, addressed not only to special
ists but also to a broader range of readers including graduate students. Topics
are advanced, while detailed exposition of ideas, restriction to typical results -
rather than the most general ones - and, last but not least, lucid proofs help to
gain the utmost degree of clarity.
It is hoped, that DYNAMICS REPORTED will be useful for those enter
ing the field and will stimulate an exchange of ideas among those working in
dynamical systems.
Christopher K.R.T Jones
U rs Kirchgraber
Hans-Otto Walther
Managing Editors
Table of Contents
Transversal Homoclinic Orbits near Elliptic Fixed Points
of Area-preserving Diffeomorphisms of the Plane
Chr. Genecand
1. Introduction................ 1
2. Elements of the Theory of Minimal States . . "_ 9
3. A Priori Lipschitz Estimates for Minimal Orbits 10
4. First Perturbation: Isolation and Hyperbolicity
of Minimal Periodic Orbits . . . . . . . . . . 14
5. Second Perturbation: Nondegeneracy of Homoclinic Orbits. 18
6. Application to Mather Sets 23
7. Special Classes of Diffeomorphisms 24
References . . . . . . . . . . . 29
Asymptotic Periodicity of Markov and Related Operators
J. K omornz'k
Introduction 31
1. Basic Notions and Results 32
2. The Reduction Procedure 40
3. Asymptotic Periodicity of Constrictive Marcov Operators 46
4. Weakly Almost Periodic Operators . . . . . . . . . . 52
5. Asymptotic Periodicity of Power Bounded Operators . . 57
6. Asymptotic Periodicity of Operators on Signed Measures 62
References . . . . . . . . . . . . . . . . . . . . 67
A Nekhoroshev-Like Theory of Classical Particle Channeling
in Perfect Crystals
H. Scott Dumas
1. Introduction................. 69
2. Background and Outline of Main Results . . . . 70
2.1 Sketch of the Development of Nekhoroshev Theory 70
2.2 Brief Description of the Physics of Particle Channeling in Crystals 74
2.3 Outline of Main Results . . . . . . . 76
3. Formulation of the Channeling Problem 77
3.1 The Perfect Crystal Model. . . . . . 77
VIII Table of Contents
3.2 The Channeling Criterion . . . . . . . 79
3.3 Transformation to Nearly Integrable Form 79
4. Construction of the Normal Forms 80
4.1 Description of Methods . . . . . . . . 81
4.2 Notation . . . . . . . . . . . . . . 82
4.3 Near Identity Canonical Transformations via the Lie Method 83
4.4 Statement of the Analytic Lemma 85
4.5 The Iterative Lemma . . . . 87
4.6 Technical Estimates .... 92
4.7 Proof of the Analytic Lemma 94
5. The Generalized Continuum Models 97
5.1 Resonant Zones and Resonant Blocks 97
5.2 Geometric Considerations . . 98
5.3 The Spatial Continuum Model 100
5.4 Channeling . . . . 105
6. Concluding Remarks 111
References . . . . . 112
The Adiabatic Invariant in Classical Mechanics
J. Henrard
Introduction ........ 117
Part I The Classical Adiabatic Invariant Theory
l. Introduction . 119
2. Action-Angle Variables 121
3. Perturbation Theory 124
4. The Adiabatic Invariant 127
5. Explicit Approach to Action-Angle Variables. 129
6. Extension of Perturbation Theory to the Case of Unbounded Period 132
Part II Transition Through a Critical Curce
l. Introduction . 134
2. Neighborhood of an Homoclinic Orbit . 136
3. The Autonomous Problem Close to the Equilibrium 138
4. The Autonomous Problem Close to the Homoclinic Orbit 142
5. Traverse from Apex to Apex 145
6. Probability of Capture 149
7. Time of Transit 153
8. Change in the Invariant 157
Part III The Paradigms
l. Introduction . 161
2. The Pendulum. 163
3. The Second Fundamental Model 167
4. The Colombo's Top . 173
5. Dissipative Forces 178
Table of Contents IX
Part IV Applications
1. Introduction . . . . . 180
2. Passage Through Resonance of a Forced Anharmonic Oscillator 182
3. Particle Motion in a Slowly Modulated Wave 186
4. The Magnetic Bottle . . . . . . . . . . 191
5. Orbit-Orbit Resonances in the Solar System 197
6. Spin-Orbit Resonance in the Solar System 204
Appendix . . . . . . . . . . . . . . . 211
Appendix 1: Variational Equations . . . . 211
Appendix 2: Fixing the Unstable Equilibrium and the Time Scale 212
Appendix 3: Mean Value of Ri('l/Ji,Ji,)...) 1 s:; i s:; 2 . . . . . . . . 214
Appendix 4: Mean Value of R3('l/J3, h,)...) . . . . . . . . . . .. 216
Appendix 5: Estimation of the Trajectory Close to the Equilibrium 218
Appendix 6: Computation of the True Time of Transit . . .. 221
Appendix 7: The Diffusion Parameter in Non-SYl.llmetric Cases. 224
Appendix 8: Remarks on the Paper "On the Generalization
of a Theorem of A. Liapounoff", by J. Moser
(Comm. P. Appl. Math. 9, 257-271, 1958) 226
References. . 230
List of Contributors 237
Transversal Homoclinic Orbits near Elliptic Fixed
Points of Area-preserving Diffeomorphisms
of the Plane
Chr. Genecand
Abstract. Extending a result due to E. Zehnder [27], we prove that generically,
an area-preserving analytic diffeomorphism of the plane has transversal homo
clinic orbits and nondegenerate Mather sets in every neighboThood of a stable
elliptic fixed point. "Generically" Tefers to a topology defined by means of the
Taylor coefficients of the mapping at the fixed point whose ordeT is higher than
a prescTibed aTbitrary integeT, all lower coefficients being held fixed. The proof
makes use of the Aubry-MatheT theory fOT monotone twist mappings.
1. Introduction
1.1 Results and Interests of this Paper. One of the most important purposes of
the theory of dynamical systems is to define and check different types of stability
for stationary and periodic solutions.
In this paper, we go in the opposite direction: starting from "stable solutions"
of 2-dimensional systems - stable in the sense of Lyapunov: nearby starting
solutions do not escape -, we prove a general result of so-to-speak "enclosing
instability", i.e. of instability in annular regions arbitrarily close to these stable
solutions.
To make such a result interesting, we restrict the problem in two ways: first
we require analyticity of the systems considered; secondly we concentrate on
such systems which are "conservative", i.e. which preserve some real quantity,
called "energy" or "Hamiltonian (function)".
Do these two quite strong restrictions not make the systems particularly
"neat" in some sense? Does, for instance, the conjunction of conservativity, ana
lyticity and Lyapunov stability not force such systems to be "locally integrable" ,
in the sense that the stable solutions had to be imbedded in a continuous foliation
of invariant cylinders?
We show that the answer is negative: such a "neat", e.g. locally integrable,
structure is quite exceptional, even for conservative, analytic, stable systems.
(Already the presence of isolated orbits prevents integrability, see Siegel [25],
Riissmann [23].)
The general case is not only not integrable, it even exhibits a very "wild"
and "chaotic" behaviour in any neighborhood of the stable solution, namely in
the so-called "regions of instability" surrounding it.
2 Chr. Genecand
We show this, in fact, not for the considered 2-dimensional systems them
selves, but, for convenience, for their time-one-maps, which are, in this case,
area-preserving analytic difJeomorphisms defined in the neighborhood of a sta
ble, say elliptic, fixed point ("elliptic" means that the eigenvalues at the fixed
point lie on the complex unit circle without the points ±1).
In this set-up, as it is well-known (see [16]), there are, generally, invariant
curves surrounding the fixed point arbitrarily closely. Our result marks an upper
limit of this famous property, inasmuch as it shows that, generally, these invari
ant curves do not form a continuous family as in the integrable case. Rather they
alternate with rings containing no invariant curve in their interior, the so-called
"rings of instability". The chaotic behaviour in one of these rings can be illus
trated by the fact that any two orbits contained in its interior come somewhere
arbitrarily close one to the other. In other words, there is a complete mixing of
the interior of such rings of instability. Such a non-integrability result for local
dynamical systems or diffeomorphisms is not new, but takes place in an already
long history beginning 100 years ago with Bruns, Poincare [20,21]' then Birkhoff
[4, 5], and culminating 50 years ago with Siegel [24, 25] and others ([15], ... ).
Here the existence of unstable regions is proven by showing existence, generi
cally, of so-called transversal homoclinic orbits. These are orbits which are
asymptotic in both directions to some hyperbolic periodic orbit (i.e. the eigenval
ues are real and =f. ±1), whose stable and unstable invariant manifolds intersect
transversally at it. See Fig. 1.
Fig. 1. Transversal Homoclinic Orbits.
In the immediate neighborhood of such orbits, the diffeomorphism actually
behaves in a "chaotic" way; in particular, there are subsystems conjugate to
so-called "Bernoulli shifts" with infinitely many symbols (see [27], § 8).
We now formulate the result more precisely, except for the analytic topol
ogy and for some slight additional condition on the eigenvalue which will be
introduced later.
Theorem. Generically (for some analytic topology), area-preserving, analytic dif
feomorphisms defined near an elliptic fixed point have transversal homoclinic
orbits in each neighborhood of this point.
Transversal Homoclinic Orbits 3
In this form, without specification of the topology, this result is not new, since
it has been proven already 20 years ago by Zehnder in [27J. However, his topology
was less fine than ours, in the following respect. To obtain existence of the
necessitated hyperbolic periodic orbits, Zehnder used a bifurcation method which
constrained him to move the eigenvalue A at the fixed point (IAI = 1, A2 =I- 1)
close to some resonant value AO = e27rip/q with rational p/q. After having done
so infinitely many times with an increasing sequence of q's (in order to obtain
periodic and homo clinic - orbits in every neighborhood of the fixed point),
it may be that one ends up with an eigenvalue Aoo = e27ria with irrational a of
Liouville type, and this could lead one to believe that the arithmetic properties
of the eigenvalue are decisive for the result.
Our main contribution consists in showing that this is not the case: to prove
the result, we need not change the eigenvalue A in any way, nor even any coeffi
cient of the Taylor expansion at the fixed point up to a fixed integer ;::: 4: only
the Taylor coefficients of higher order are changed, and as little as one wants.
As a consequence, for ~ given concrete diffeomorphism,the property of having
transversal homoclinic orbits arbitrarily close to the fixed point, and thus also the
integrability resp. non-integrability of this diffeomorphism, can not be decided
if merely a finite number of its Taylor coefficients at the fixed point are known:
this information depends always on the knowledge of the whole Taylor expansion.
This is the main point cleared up in this paper.
Further interests of this paper lie, besides in the result itself, - and perhaps
even more - in the methods used to prove it, which are roughly described now.
First, the mentioned refinement of Zehnder's result is made possible by the use
of a quite recent theory which solves the question of existence of the desired
periodic and homoclinic orbits. This is the so-called theory of minimal states
or orbits, developed simultaneously and independently by Aubry and Mather in
the early eighties. It allows us to obtain the desired orbits as "minimal orbits"
for a simple variational principle coming from the solid state physics, which will
be exposed in Sect. 2.
The difficulty in using this theory is that it is a global theory, while our prob
lem is of local nature. It is therefore necessary to prove some a priori Lipschitz
estimates for minimal orbits, which allow then to localize the latter into some
annuli which are relevant for us. These Lipschitz estimates are derived in Sect. 3,
and they are of interest by themselves.
After having obtained, in this way,' existence of the desired orbits, we still
have to make them nondegenerate, i.e. to make the periodic orbit hyperbolic,
and then the homo clinic orbit transversal. This is done essentially by means of
elementary perturbation theory, and it necessitates two formulae having their
own interest, namely:
- for the first perturbation, which makes the minimal periodic orbit to a hy
perbolic minimal orbit: a "monotone correspondence formula for generating
functions", relating, in a monotone way, perturbations of two different so
called "generating functions" one the other. Generating functions will be
defined in 1.2 and in Sects. 2 and 3. See Sect. 4 for this formula.