Chaos in Gravitational N-Body Systems: Proceedings of a Workshop held at La Plata (Argentina), July 31 – August 3, 1995

313 Pages·1996·13.74 MB·English

by  G. Contopoulos

CHAOS IN ORA VITATIONAL N-BODY SYSTEMS
CHAOS IN ORA VITATIONAL
N-BODY SYSTEMS
Proceedings of a Workshop held at La Plata (Argentina),
July 31-August 3,1995
Edited by
J. C. MUZZIO
Observatorio Astron6mico, Universidad Nacional de La Plata,
La Plata, Argentina
S. FERRAZ-MELLO
Instituto Astronamico e Geofisico, Universidade de Silo Paulo,
Silo Paulo, Brazil
and
J.HENRARD
Departement de Mathematique, FNDP, Namur, Belgium
Partly reprinted from Celestial Mechanics and Dynamical Astronomy,
Volume 64, No.1, 1996
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging-in-Publication Data
Chaos in gravitational N-body systems: a workshop held at La Plata,
Argentina, July 31-August 3, 1995 I edited by J.C. Muzzio, S. Ferraz
-Mello and J. Henrard.
p. cm.
1. Many-body problem--Congresses. 2. Celestial mechanics
-Congresses. 3. Chaotic behavior in systems--Congresses.
I. Muzzio, Juan C. II. Ferraz-Mello, Sylvia. III. Henrard, J.
OB362.M3C47 1996
521--dc20 96-28459
ISBN-13: 978-94-010-6623-5 e-ISBN-13: 978-94-009-0307-4
DOl: 10.1007/978-94-009-0307-4
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The ~therlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk: and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved
© 1996 Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
PREFACE vii
G. CONTOPOULOS and N. VOGLIS / Spectra of stretching numbers and helicity angles in dynam
ical systems 1
C. FROESCHLE and E. LEGA / On the measure of the structure around the last KAM torus before
and after its break-up 21
R.H. MILLER / Some comments on numerical methods for chaos problems 33
J.A. NUNEZ, P.M. CINCOTTA and F.C. WACHLIN / Information entropy 43
D. MERRITT / Chaos and elliptical galaxies 55
J .C. MUZZIO / Sinking, tidally stripped, galactic satellites 69
A. BRUNINI / On the satellite capture problem 79
S. FERRAZ-MELLO, J.C. KLAFKE;, T.A. MICHTCHENKO and D. NESVORNY' / Chaotic transi
tions in resonant asteroidal dynamics 93
J. HENRARD / Structure of the phase-space in the 3D problem of the 2/1 and the 3/2 Jupiter reso
nance 107
J. LASKAR / Large scale chaos and marginal stability in the solar system 115
M. LECAR / Chaos in the solar system 163
P. CIPRIANI, G. PUCACCO, D. BOCCALETTI and M. DI BARI / Geometrodynamics, chaos and
statistical behaviour of N-body systems 167
D. BOCCALETTI, M. DI BARI, P. CIPRIANI and G. PUCACCO / Geometrodynamics on Finsler
spaces 173
A.A. EL ZANT / Stability of motion of N-body systems 179
T. TSUCHIYA, N. GOUDA and T. KONISHI / Microscopic dynamics of one-dimensional self-gravi
tational many-body systems 185
A. BRUNINI; C.M. GIORDANO and A.R. PLASTINO / Numerical exploration of the dynamics of
self-adjoint S-type Riemann ellipsoids 191
A.M. FRIDMAN and O.V. KHORUZHII / On the possibility of 2D dynamical chaos in astrophysical
disks 197
vi
A.M. FRIDMAN and O.V. KHORUZHII / The observed turbulent spectrum of cloudy population of
the Milky Way as an evidence of a weak Rossby wave turbulence 207
P.E. ZADUNAISKY and C. FILICI / On the accuracy in the numerical treatment of some chaotic
problems 209
O.C WINTER and C.D. MURRAY / The Liapunov exponent as a tool for exploring phase space 215
P.M. CINCOTTA, A. HELMI, M. ME'NDEZ and J. NU'N EZ / Information entropy as a tool for
searching periodicity 221
A. HELMI and H. VUCETICH / Painleve analysis of simple Kaluza-Klein models 227
S. BLANCO and O.A. ROSSO / Nonlinear analysis of a classical cosmological model 233
C.D. EL HASI / Nontrivial dynamics and inflation 239
N. VOGLIS and C. EFTHYMIOPOULOS / A model distribution function for violently relaxed N
body systems 245
Y. PAPAPHILIPPOU and J. LASKAR / Global dynamics of the logarithmic galactic potential by
means of frequency map analysis 253
A. MEZA and N. ZAMORANO / Numerical stability of Osipkov-Merritt models 259
L.P. BASSINO, J.C. MUZZIO and F.C. WACHLIN / Structure of galactic satellites 265
S.A. CORA, J.C. MUZZIO and M.M. VERGNE / Dynamical friction effects on sinking satellites 271
T. FUKUSHIGE / Gravitational scattering experiments in infinite homogeneous N-body systems 279
H. UMEHARA and K. TANIKAWA / Dominant roles of binary and triple collisions in the free-fall
three-body disruption 285
M.G. PARISI and A. BRUNINI / Dynamical consequences of Uranus' great collision 291
E. KOKUBO and S. IDA / Runaway growth of planetesimals 297
D. NESVORNY / Secondary resonances inside the 2/1 Jovian commensurability 303
T.A. MICHTCHENKO / Applications of Fourier and wavelet analyses to the resonant asteroidal mo
tion 307
1.1. SHEVCHENKO / Spectra of winding numbers of chaotic asteroidal motion 311
PREFACE
The Workshop on Chaos in Gravitational N - Body Systems was
held in La Plata, Argentina, from July 31 through August 3, 1995. The
School of Astronomy and Geophysics of La Plata National University, best
known as La Plata Observatory, was the host institution. The Observatory
(cover photo) was founded in 1883, and it has nowadays about 120 faculty
members and 70 non-faculty members devoted to teaching and research in
different areas of astronomy and geophysics.
It was very nice to see how many people, from young students to
well recognized authorities in the field, came to participate in the meeting.
This audience success was due to the increasing understanding of the neces
sity to gather together people from Celestial Mechanics and Stellar Dynamics
to explore the problems that exist at the frontier of these two disciplines and
their common interest in chaotic phenomena and integrability (the famous
Argentine beef was, certainly, also an attraction!).
All the papers of the present volume were refereed. Most were
accepted after some revision, while some needed no change at all (compli
ments to their authors!) and, sadly, a few could not be included. About half
a dozen authors did not submit their contributions for publication, mainly
because they were already in print elsewhere. Therefore, the special issue
of Celestial Mechanics and Dynamical Astronomy includes all the invited
lectures of the workshop, while the proceedings volume includes those same
lectures plus the bulk of, but Bot all, the contributions to the meeting.
The Scientific Organizing Committee was formed by S. Ferraz
Mello, C. Froeschle, J. Henrard, J. Laskar, M. Lecar, J. Makino, G. Meylan
and J.C. Muzzio (Chairman), while A. Brunini, P.M. Cincotta, C.M. Gior
dano, J.A. Nunez (Chairman), M.M. Vergne, H.R. Viturro and F.C. Wachlin
made up the Local Organizing Committee. We are very grateful to all of
them, as well as to the La Plata students 1. Chajet, J. Douglas, E. Gularte,
1. Mammana, 0.1. Miloni, A. Torres and R. Vallverdu, faculty member S.A.
Cora, and non-faculty members S.D. Abal de Rocha, M.C. Fanjul de Cor
rebo, M.C. Visintfn and G. Sierra who were all of great help. The meeting
was sponsored by the Organization of American States and by the School of
Astronomy and Geophysics of La Plata University, and it was supported by
vii
viii
the generous contribution of La Plata National University, the International
Centre for Theoretical Physics, the Photometry and Galactic Structure Pro
gram of the National Research Council of Argentina (CONICET), and the
Latin - American Center of Physics. Our warmest thanks to all of them.
J.C. Muzzio S. Ferraz-Mello J. Henrard
SPECTRA OF STRETCHING NUMBERS AND HELICITY
ANGLES IN DYNAMICAL SYSTEMS
G. CONTOPOULOS and N. VOGLIS
Department of Astronomy, University of Athens
Panepistimiopolis, GR 15784 Athens, Greece
Abstract. We define a "stretching number" (or "Lyapunov characteristic number for
one period") (or "stretching number") a = In I e~~l I as the logarithm of the ratio
of deviations from a given orbit at times t and t + 1. Similarly we define a "helicity
et
angle" as the angle between the deviation and a fixed direction. The distributions of
the stretching numbers and helicity angles (spectra) are invariant with respect to initial
conditions in a connected chaotic domain. We study such spectra in conservative and
dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees
of freedom. In 2-D conservative systems we found that the lines of constant stretching
number have a fractal form.
Key words: Lyapunov characteristic numbers - stretching numbers - helicity angles -
conservative and dissipative mappings
1. Stretching Numbers
The maximal Lyapunov characteristic number (LCN)
et
LC N = lim ~ In I eo I (1)
t-+oo t
eo, et
(where are the infinitesimal deviations from a given orbit at times 0
and t respectively) provides a good characterization of chaos. The LCN is
zero in ordered regions while it is positive in chaotic regions. However in
practice the limiting value of LCN is reached after a long time that may be
unrealistic. E.g. in galactic dynamics in order to obtain good results we need
in general t of the order of 105 - 106 periods, while the age of the Universe
is of the order of 102 periods.
In recent years there has been much interest in finite time Lyapunov
numbers (e.g. Fujisaka 1983, Grassberger and Procaccia 1984, Benzi et al.
1985, Udry and Phenniger 1988, Sepulveda et al 1989, Ababarnel et al.
1992). Most of the information about a dynamical system is provided when
t is small, rather than large. In mappings the smallest t is 1 period. The
I-period Lyapunov characteristic number was introduced by Froeschle et al.
(1993) and by Lohinger et al. . (1993), and was studied in detail by Voglis
and Contopoulos (1994) both for conservative and dissipative mappings.
The quantity
(2)
Celestial Mechanics and Dynamical Astronomy 64: 1-20, 1996.
@ 1996 Kluwer Academic Publishers.
2 G. CONTOPOULOS AND N. VOGLIS
5". ~7.0. t.CN~'.27e
7..0-: J.0'- .L.. ... -~2~.-0-' .. .·..I1o.L. 0 l.o0t!.0 J'.l1.I.0, 2.0 :J.0' l.a
a) (b)
B
1.5 1.5
0.
1.10 1.0
0.S e.'>
2
0 0.0-J .0 -2,0 -, .~ 0.0 1,0 '" .0
H. ... f.-5J21. LCH-'~~S
-J 0 -2.0 -1.0 0.0 '.0 2.0 3.0
2.0 ""II."""J _ •• I .. .I_"-,l-'-L--'...I l .. , •• 4l."
) (d)
1.6 1.6
1.0 1.0
0.6 €I.E
0.0 0.0
-3.0 3.
Figure 1. (a) The distribution of 5 x 104 consequents in the standard map for K = 7,
and initial conditions (x = O.l,y = 0.5,e,,, = l,~y = 0). (c) The same distribution in
the Henon conservative map for K = 5.321(b = 1) and the same initial conditions. The
corresponding spectra are shown in (b) and (d) respectively. The Lyapunov characteristic
numbers in both cases are equal LeN =1.276, but the spectra are completely different.
is called a stretching number. The distribution of the stretching numbers
(spectrum) gives the fractional number dff of values of a in an interval
+
(a, a da), divided by da:
!::J.N
(3)
S(a) = Nda
as a function of a.
The spectrum gives more information about a system than the Lyapunov
characteristic number, or the distribution of the consequents on a Poincare
SPECTRA OF STRETCHING NUMBERS AND HELICITY ANGLES 3
surface of section. In Fig.1 we compare the spectra of two different mappings,
that have the same LCN. The first spectrum (Fig. 1b ) corresponds to the
standard map
1(.
Xi+! = Xi + Yi+! Yi+! = Yi + 27r sm 27rXi (modI), (4)
while the second spectrum (Fig. Id) corresponds to the conservative Henon
map
xr -
=
_ Xi+! 1 - 1(' Yi (modI), (5)
with b = 1.
The corresponding distributions of the consequents on the plane (x, y) are
given in Figs. 1a and 1c. The values 1( = 7 and 1(' = 5.321 were chosen in
such a way that the Lyapunov numbers are equal LCN=J.276. Furthermore
both systems look completely chaotic (in fact there are some very small
islands that are not seen in Figs. Ia and Ic). But although the two usual
criteria of chaos (LCN and distribution of consequents) are the same, the
spectra are quite different. This means that there are basic differences in the
structure of these two systems.
The LCN is the average value of the stretching numbers ai.
The usefulness of the spectra is based on the fact that the spectra are
invariant (a) with respect to the initial conditions along the same orbit, (b)
eo,
with respect to the direction of the initial deviation and, more important,
(c) with respect to initial conditions in a connected chaotic domain (in the
case of chaotic orbits).
In the case of ordered orbit the last invariance is replaced by (c') invari
ance with respect to initial conditions along the same invariant curve (or
torus). But the spectra of various invariant curves are different. Further
more, as we see in. Fig. 1, the spectra of different systems are also different.
2. Helicity Angles
The stretching numbers give much information about a dynamical system,
but an equally important information is provided by the "helicity angles"
ei
namely the angles </>i of the vectors with a given direction. These angles
are definedrin the interval (-180°,180°).
The name "helicity angle" was chosen because in more degrees of freedom
ei
the vectors are not in the same plane.
The spectrum of the helicity angles is defined in the same way as the
spectrum of the stretching numbers. Namely S( </» is the fractional number
~ of the helicity angles in the interval (</>, </> + d</», divided by d</>.