Recurrences and Discrete Dynamic Systems
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809 Igor Gumowski
Christian Mira
Recurrences and Discrete Dynamic Systems
Springer-venag Berlin Heidelberg New York 1980
Authors Igor Gumowski Christian Mira U.E.R. de Mathematiques, Universite Paul Sabatier, 118, Route de Narbonne 31077 Toulouse France
AMS Subject Classifications (1980): Primary: 58-02 Secondary: 26A18, 34C15, 34C35, 39A15, 58 F05, 58F10, 58 F15, 58 F20, 58F99 ISBN 3-540-10017-2 ISBN 0-387-10017-2
Springer-Verlag Berlin Heidelberg NewYork Springer-Verlag NewYork Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under c954 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. ©.by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
The idea to study recurrences of P. Montel. A discussion
took root in 1949 during an informal
about possible
types of fixed points
lasting collaboration with C. Mira. The analytical
in 1959 led to a
insight increased more rapidly
after an invariant curve germ, based on a Latt~s
series, was successfully
by means of numerical
described
computations.
The material
tutes a synopsis of the slowly accumulated with R. Thibault, G. Targonski
R. Clerc, Ch. Hartmann,
produced
substantial
particular
continued
in this monograph
consti-
results. Frequent discussions
J. Couot, C. Gillot, O. R~ssler and
improvments.
During the last few years recurrences science and they have become a major research Systems Research Group of Toulouse University. have been impossible without
lecture
the continued
appeared
in several fields of applied
topic of the interdisciplinary
Dynamic
The creation of such a group would
support of J.C. Martin, President of
Toulouse University. The text was typed by Mrs. C. Grima and many of the figures were drawn by G. Roussel.
The preparation
contributions
of the typescript was encouraged by Ph. Leturcq.
are gratefully acknowledged.
Toulouse, March
1980.
I.G.
All
CONTENTS p°
Introduction
I
and st a,tement of the problem
Chapter I - Some properties of first order, recurrences I.O
Introduction
I.|
Elementary properties
1.2
Singularity
1.3
Relation
1.4
Topological
23 of first order recurrences
structure of a quadratic
to functional
29
recurrence
38
equations
46
entropy and some statistical
Chapter II - Some properties
properties
57
of second order recurrences
2.0
Introduction
2,!
Elementary
2.2
The effect of non linear perturbations
76
2.3
Influence
81
2.4
Approximate
2.5
Some relationships between recurrences, differential equations
61
properties
of linear recurrences
65
domain of a stable point singularity d
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