Iterated Maps on the Interval as Dynamical Systems 9780817649272

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Iterated Maps on the Interval as Dynamical Systems

Pierre Collet Jean-Pierre Eckmann

Reprint of the 1980 Edition

Birkhauser Boston • Basel • Berlin

Pierre Collet Centre de Physique Theorique Ecole Polytechnique 91128 Palaiseau Cedex, France collet@ cphl. pol ytechnique. fr

Jean-Pierre Eckmann Departement de Physique Theorique Universite de Geneve 1211 Geneve 4, Switzerland jean-pierre.eckmann@ physics. unige.ch

Originally published in the series Progress in Physics

ISBN 978-0-8176-4926-5 00110.1007/978-0-8176-4927-2

e-ISBN 978-0-8176-4927-2

Library of Congress Control Number: 2009932682 Mathematics Subject Classification (2!XXl): 26A 18, 37-XX, 37E05, 46T20

© Birkhauser Boston, a part of Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer Science+Business Media. LLC. 233 Spring Street, New York, NY. HXll3, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software. or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms. even if they are not identified as such. is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkhauser Boston is part of Springer Science+ Business Media (www.birkhauser.com)

PPh1 Progress in Physics Edited by A. Jaffe and D. Ruelle

Iterated Maps on the Interval as Dynamical Systems Pierre Collet Jean-Pierre Eckmann

Birkhauser Boston • Basel • Berlin

Jean-Pierre Eckmann Departement de Physique Theorique Universite de Gen~ve CP-1211 Gen~ve4 Switzerland

Pierre Collet Ecole Polytechnique Palaiseau France

Library of Congress Cataloglng-ln-PubUcatlon Data Collet, Pierre, 1948Iterated maps on the interval as dynamical systems. (Progress in physics ; 1) Bibliography: p. Includes index. 1. Differentiable dynamical systems. 2. Mappings (Mathematics) I. Eckmann, Jean Pierre, joint author. II. Title. m. Series: Progress in physics (Boston); 1. QA614.8.C64 003 80-20751 ISBN 3-7643-3026-0

CIP- Kurztltelaufnahme der Deutschen BibUothek Collett, Pierre: Iterated maps on the interval as dynamical systems I Pierre Collet : Jean-Pierre Eckmann. -Boston, Basel, Berlin : Birkbliuser, 1980. (Progress in physics : 1) ISBN 3-7643-3026-0 NE: Eckmann, Jean-Pierre: Printed on acid-free paper 10 1980 Birkbliuser Boston 2nd printing, 1981 3rd printing, 1983

f)

Birkhiiuser

}J

4th printing, 1986 5th printing, 1997

Copyright is not claimed for works of U.S. Government employees. All rights reserved. No partofthis publication may be reproduced, stored in aretrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkbiiuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkbil.user Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-3026-0 ISBN 3-7643-3026-0 Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in USA

9 8 7 6 5

TABLE OF CONTENTS INTRODUCTION PART I.

IX

MOTIVATION AND INTERPRETATION

1. One-parameter families of maps

2. Typical behavior for one map 3. Parameter dependence 4. Systematics of the stable periods 5. On the relative frequency of periodic and aperiodic behavior 6. Scaling and related predictions 7. Higher dimensional systems

1 7 23 27

30 36 56

MATHEMATICAL ASPECTS AND PROOFS PART II. 1. 2. 3. 4. 5. 6. 7. 8.

PROPERTIES OF INDIVIDUAL MAPS

Unimodal maps and their itineraries The calculus of itineraries Itineraries and orbits Negative Schwarzian derivative Homtervals Topological conjugacy Sensitive dependence on initial conditions Ergodic properties Entropy

63 71

83 94 107 122 135 149 168

PART III. PROPERTIES OF ONE-PARAMETER FAMILIES OF MAPS 1. 2. 3. 4.

One-parameter families of maps Abundance of aperiodic behavior Universal scaling Multidimensional maps

173 184 199 227

REFERENCES

239

INDEX

245

LIST OF MATHEMATICAL SYMBOLS

248

VIII

Relations between the sections o) , 0

or

=

=

with

which is of the desired form. The other method can be applied if Eq. (1) has a periodic solution, i.e., if for some initial condition x (0) = x 0 we find x (T) = x ( 0) for some T > 0. Then we consider a hyperplane mn-1 of dimension n-1 transverse to the curve t-+x(t) through x 0 , and in this hyperplane a (small) neighborhood C¥1 of

Figure I.l.

3

Then a map

P: iW + 1Rn-l

is induced by associating to

the next intersection with initial condition occurs at to go from

y

1

y

x(O)

= y.

to

y

of the trajectory with

If the first such intersection

Py = y

we define 1

1Rn-l

y EIW

1

•

(Note that the time needed

may be different from

T, and some

points might never return, a case we do not consider.)

P

is

then a map IW+1Rn-l, and this is how flows reduce sometimes to maps.

Given a map

P: 1Rn-l + IRn-l,

it may still be possible

to reduce the discussion further (at least in part) if the map p

leaves invariant a set of "parallel" hypersurfaces of

dimension

n-2

in

1R

n-1

•

That is, it maps one hyperplane

w'

PW'

w

PW Figure I.2. onto another hyperplane.

Using each plane as a coordinate,

we are reduced to a one-dimensional problem. about this one-dimensional problem or about

Information P

will give us

some insight about the behavior of the flow itself. 3.

In many applications of natural sciences, the dynamical

system depends on controllable quantities, which we call parameters for short.

Examples are temperature, Reynolds number

or energy fed into a system.

A system may depend on many

parameters, but we shall see that in fact every one-parameter family of one-dimensional maps of a suitable type will go through the whole spectrum of possibilities.

Therefore, in

one dimension it is sufficient to study one-parameter families of maps, since no new phenomena can be found by using more parameters.

4. If we are given a one-parameter family of maps, parametrized by 1.1 E lR there is still another variable quantity, namely the choice of initial point x 0 E [-1,1]. We shall be interested in the behavior of a "typical" initial point for a "typical" value of the parameter. Of course, appropriate notions will be developed below. But even in this informal introduction we want to stress the importance of looking for typical behavior. As we shall see, even for very well behaved maps there may be some "atypical" initial points for which something chaotic happens, but such initial points are extremely rare in the measure theoretic sense. In particular the title "Period 3 implies chaos " Li-Yorke [1975] has, in the past, led to some confusion about this matter. 5. The discussion of properties of maps on the interval is extremely streamlined by some simplifying assumptions of which we now mention a few. We will indicate whenever extensions of results apply without these assumptions. This review only deals with continuous maps of the interval [-1,1] to itself. The results connected with the natural ordering of the real numbers break down when the continuity is abandoned. The choice of the interval [-1,1] is of course arbitrary, since the change of coordinates x= (2x'-(b+a))/(b-a) will transform a map [a,b]-> [a,b) into a map [-1,1)-> [-1,1). If a=-"' and/or b = +"' we are in a truly different situation, which can however often be reduced to the situation [-1,1) by the following type of discussion. Assume, e.g., a=O, b=co and f(x) ->c with leiO f(x), sup X->O f(x)) = [a',b') and we are reduced to the case of finite [a,b) if we consider instead of initial points x E [0,"'] their images f (x) as initial points. On the other hand, if f (x) tends to infinity as x-> "'• there may be a set of points U, such that fn (x) ->"' as n->"' for all x in u. Thus U is the set of points which escape to infinity. It may happen that the complement of U is an

5

interval, and in this case we may be reduced to the previous discussion.

f

0

Figure 1.3. We shall consider only maps with one maximum and we assume that this maximum occurs for x = 0. This can sometimes be achieved by a differentiable coordinate transformation. We further assume f is monotonically increasing for x < 0, and monotonically decreasing for x > 0, and once continuously differentiable (see II.l, for details). A very useful simplifying assumption which we shall make throughout is the convexity statement of "negative Schwarzian derivative,a namely that If' 1-l/ 2 is a convex function on x < 0 and on X> 0 (cf. II.4).

6

f(x)

= 1-

/

Figure !.4.

A t¥Pica1 function

If 1-1/2. I

f, and the graph of

1.4x 2

I.2

TYPICAL BEHAVIOR FOR ONE MAP

Before we study parametrized to analyze individual maps.

families of maps, we want

We are interested in the possible

behavior of the successive images of an initial point the interval

[-1,1]

for

a fixed

map

f.

x

on 0 For this we first

outline a graphical method for determining the iterates xn = f n (x ). Here, we define f n (x ) = f(f n-1 (x )). The 0 0 0 following Figure I.S shows how this is done through the rule: Go from

x

0

to the graph of the function, from the graph to

the diagaonal, from the diagonal to the graph, • . . .

Figure I . 5.

f (x) = 1 - 1. 4 x

2

-

- = f n (x) does not move at all; a fixed J20int of f. Physically speaking, this means that i f the system is at X at some Note that the point marked f(x> = x. We shall call X

-

-

X

-

P. Coller and J.-P. Eckmann. Iterated Map.'"" till' lnterralm Dmamical S_n"ll'm.>. Modern Birkhiiuscr Cla,sic,. DOl I 0.1007!97X-O-X 176--t927 -2_2. 1, then there is a neighborhood ~

x

x.

11

of

-

x

when

such that

f

no

(x 0 )

'=-l 1 0 0 and then x = -1 for all n