A GENERALIZATION OF THE LOGISTIC EQUATION

b y

Valérie Poulin

B.%. Université de Sherbrooke

A thesis siibniittecl to

t lie i k i i l r ~ of Ci racliiate St iiclies and Research

in part id fdfillrnetit of

t tic recliiiremerits for the tlegree of

,\las ter of Science

Carleton Lhiversity

Ott aiva. Ontario

Sovember, 1999

@ 1999. ValPrie Poulin

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Abstract

Tuo families of non-conjugate E~inctioris whicli generalize the well-knoxn logistic map

. = Xr,(l - x,) for X = -1 are defined. ancl sonie of t.heir biisic properties stiiclied.

I t is sliown that the Lyapiiriov esponent of al1 the mnps contained in t liese families cnn

be espressecl by a simple formula and t hat. despi te t lie dissiniilarity of t lieir polynornizil

cspressions. they al1 possess the same invariant density. The Lyapiinov esponents are

calculatecl by constriicting the conjiigate maps of the fiinctions and are shown to be

positive whicli is a strong inclicatiori t liat dl the fiinct ions are chaotic. Tlic ctinoticity

properties arc fint her st iicliecl iisirig several otlier trict hocls. '\ioreowr. t iv t ~ o faniilies

of polynominl niaps are bridly Iml;ed ;it Eroni an algcbriiic point of vien-.

hcknowledgements

1 no~ilcl like to thank Dr. .-\. .\Iingarelli for the \varni ancl receptive manrier in wliicli

lie acceptecl to bc part of this project. as mj* s~ipervisor. W i t h Iiis guidance. 1 have gainec!

miiturity. self-conficlence aiid knowleclge. niore reiisoris to be grateful to hini t o d e .

1 ~visli to espress rny appre~iat~ion to Dr. D. Offiii ancl Dr. L. May for having taken the

t inie to rcacl ni>* tliesis and for hiiving been part of the final process. Their observations

were signifi cnnt and. in a short t ime fraine. t 11- managed to give nie greiit encouragement.

1 ~ o u l c l l i ke to acknowledge t lie Sat ional Sciences and Engineering Researcli Coiincil

of Cariatla (SSERC). The trust tliat, th- have slioivn nie iws a very contributory Factor

in niaking this researcli possible.

The expression of ni? grarit i d e u-oiilcl not be complete. were L not to include my good

friencl. Hugo Touchet te. For his nioral support t hroughoii t t his project. his indispensable

and aciite aclvice and for making Iiimself available at al1 time.

Contents

1 Introduction 1

3 General Definitions 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Orbits 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 ChaoticBehavior G

. . . . . . . . . . . . . . . . . . . . . . . . . . . . X3 Probnbiiist ic .-1 pproacli 10

3 Logistic Family 14

. . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definit ion of the Logistic Alap 1-1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Bifurcation Diagram 1.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . 3 Lyapiinov Esponents 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 / \=4.EsactRest i l t s 19

4 Generalization of the Logistic Family 22

. . . . . . . . . . . . . . . . 4.1 Definition of two Sets of Fiinctions: S and C 22

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Particularities of S and C 28

. . . . . . . . . . . . . . . . . . 1.3 Conjugate Llaps of fiinctions in S and C 34

. . . . . . . . . . . . . . . . . 4.4 Lyapiinov Esponents of S X ( x ) and CA'(x) 39

5 Conclusion

Chapter 1

Introduction

Tlie logistic inup was oripinally of interest to the ecologists ~ i i o wanted to mode1 the

popiilatiori of species. To siniplify the problem, they tried to mode1 the worlcl in ternis

of cliàr-rete tinie intervals. One of the simplest moclels proposed (Nayniird Smith 1968)

\vas t lie following:

.~n+l = /\:rn(L - xn)

non- known as the logistic equutiori. In tiiis nioclel ro is the initial population. .cn is the

popiilat ion n years \or an!- t inie iiitermls) liiter and the X-valiie represents tlie reprodiic-

tion rate of tlie st~icliecl population. The large Lucination for this fiinction is esplaineci

by the effect of the X-value on the mode1 itself: for various values of X. the mode1 be-

liaves conipletely differently. At X = 4. for instance. one can get a formula to espress the

popdation at any time in terms of n: z, = f (n).

Sowadqs. the logistic equation is probably the most popular esample found in the

Dynamical Systems and Chaos literature. The. reason for it. is that it is a very simple

fiinction which arises nnturally in man' contests and it has a wide variety of t - pes of

dynamics.

In this thesis. we generalize the logistic fiinction to two sets of fiinctions S and C t hat

exhibit very similar beliavior. \\'e begin in Chapter 2 kg. giving the necessary terrninology

and by introdiicing various concepts r liat are stuclied in the following chûpters: sensitive

depenclence on initial conditions. Lyiipiiiiov esponent. bifiircatiori diagram. invariant

nieLisure. etc.

Cliapter 3 is devoted to the st~idy of the logistic fiiiiction. \\é s t~idy its chaotic

beliavios by analyzing. tliroiigli clifferent approiicties, several key propcrties of chaos.

Every resiilt presentecl in tliis chiipter can be foiind in basic literatwe [S. 17: 141, its

principal sole is to ~iiotimte tlic observations made un Functions ol S nncl C in the nest

cliapter.

Finnlly. Cliapter -1 delils \vit l i tlic geiieralizat ion itself. I t contiiiris t lie definitions

of t lie srts S aritl C aricl i t ~iiiclcrli~ics tlic sirniliirities betumti tlic stticly of the logistic

fiinctiori and the functions iri S and in C. .\Ioreover. ive brietly clescribe ari iilgebraic

property of tliese two sets ancf u-e make some connections bctwecn clements of S and

elerrients of C.

Chapter 2

General Definitions

in this chapter. WC iiit rocluce the basic clefinitions and resiilts relevant to the s t iidy of

the logistic fiinction. Thesc definitions ran be fourid in stanciard testbooks on "Chaos

tlieory" or "Discrete Dynamical SJ-stenis" . [5. 12. 141.

2.1 Orbits

In tliis section. ww consicler a niap f : I - I where I c 8. \\é s q that cvc iter-ate J

when ive ewliiate it over and o\.er iising the outp~it of the previous application as the

input for the nest: f (...(f(r))...). or if / is composed witli itself a number of tirnes.

Definition 1 .4 dynamical system consists o / a set of possible states. together with n

rvle that d e t e m i n e s the present state in terms ofpast states. Ré denote the initial state

by q. The state changes urith tirne under the rule x,+l = f (x,). Here. the uariable n

stands for time. and I, designates the date ut time n.

Defnition 2 Gioen .ro E W . we define the orbit of xo under f to be the sequence of

&rations o f f . {zo f (zo), f ? ( ~ ~ ) ~ ...}. Ive use the following notation: x. = f "(xo).

Definition 3 IVe cal1 p a periodic point of period b if fqP) = p . and if k is the

smallest such positiue integer. The orbit uith initial point p [u~hich cunsists 01 k points)

is called a periodic orbit of period k . IVhm A. = 1 i.e.. f ( p ) = p. ,tue suy that p is a

fixed point of the .map f. The orbit {rot xi, ...) is eventually periodic if there exists

an integer !Y for u~hich {rs. rLv,l, ...} is a penodic orbit. Finallg. the orbit {JO. xi. . . .)

is said to be asymptotically periodic if there exists u periodic orbit {yo. y,. ...} srrch

that lin1 12, - y,[ = 0. 11-32

Tliese are niairiiy two different types of fised points: attractiny and repelling fised

points. also callecl s i n b and sources.

Definition 4 Let p be a k e d point. IVe say thnt p is a sink (a t t rx t ing fiseci point) if

there is an e > O such that j ir ail x in the E-neighborhood iVE(p) ,, lim f ' ( : r ) = p. lVe cul1 k-,32

p a source (repelling fisecl point) ij there is an s-neighborhood X ( p ) s,ucit th.at each 2

Soie tliat the above definitions can also be iised to characterize periodic points (a t -

tracting ancl repelli-ng periodic points) since a k-periotlic point of f is a fisecl point of fk.

An attracting periodic point of / is a sink of f<ancl a repelling periodic point of f is a

source OF fk.

Theorem 2.1.1 Let f be a contincwus function for u.ihich derivatiues of al1 orders exist

(Le., f is a smooth function). and assume that p is a fized point off.

1. If lf'(p)l < 1: then p is a sink.

2. If 1 ft(p)l > 1 , then p is a source.

ProoE See [14: p.1Oo Theorem 1.5).

Example 1 Let

2, if O sr 5 112

2(1 - z): if 112 5 :r 5 1

This map is usually called the tent rnap. In Figwe 1: u e c m see that f has only two

Fxed points: O and 213. Those fixed points are sources since 1 f'(x)l = 2 > 1 for* al1 x.

Figure 1 Fised points of t.iie terit niap f(z): / ( s ) = r nt O and 2/3.

The poirrts O und 213 ur.e sho+un in F i p r e 2 since they satisjg the eyrration j2(z) = I.

The other tuw points 215 und -11.5 ore thus period-t i~~o points.

Figure 2 Fised points and period-two points of the tent map:

f2(z) = x at 0.2/5.2/3 and 415.

Notice that every periodic point of j is a repelling periodic point since l(fk)'(x)! = 2 5 1

for al1 positive integels k and for al1 points x for uhich the den'.c~ati.ce o f f "ezsts.

Lemnia 2.1.1 If j(jk)'(x)J > I for all positive integers k and for al1 points 2 for which

the derivatiue of fk exists thert f does not haue any asymptotically periodic orbit other

than ezient,ually periodic orbits.

ProoE Let the orbit (xo. .LI. ...} be an ~isyniproticall>f periodic. urbit uf /. Le.. tliere

esists 3 peri~clic orbit of peiiod k. {go. gi , ...} siicli tliat lini 12, - y,l = O. r o t e tliat n - x

{go: yl;, MI. ..) is ii constant secpence ancl lin1 s,l; = y0 whicli implics that for dl r > 0. n-iu

t tiere esists aii integer 3 sucli t hat 1 (f 7 )"x0) -go [ < e for al1 n > Ar. But y0 is a repelling

fisecl poirit of f%ince I(/")'(yo)l > 1. Le.. there esists E > O siich tliat each .r E :\'.(yo)

cscept for go itsclf eventiially niaps outside of - (go) wlien )ire iterate f \ This implics

tliat ( fk)"( .ro) = yu foi d l n > :Y ancl so {ro. q. x?. ...} is eventiiiilly perioclic. CI

2.2 Chaotic Behavior

.\laps of the real lirie are sait1 to have chnotic bcimpior when t l i q have chaotic orbits.

.As for a soiirce, an orbit t h t is diiiotic is one tliat eshibits an iinstable beliavior but

t hat is not itself fisecl or periodic. C haot icity is generally a ver!* complicatecl property to

verif'.. Because of similaritics between the orbits of a map and those of a map's conjugate.

conjugacies are iiseftil in characterizing the behavior of sonie fitnctions under iteration.

Definition 5 Let J : I - I and g : J + J be t v o rnaps. CVe say that f and g are

conjugate ij them is a horneomo~rphism El : I -+ J (bijectiue and contin,uow nlap). such

that H 0 / = g 0 H. The function H is called a conjugacy.

=in interesting fact is that conjiigac~- is an equin~lence relation on the set of functions

of the form j : I -, I . It is easy to show that wlien xo is a periodic point of f of period

k . then H(xo) is a periodic point of g of period k .

Lemma 2.2.1 Let f, g be tu10 conjugate maps mith f o H = H O g . L,et {ru, XI. r?! ...) be

an orbit of g. Shen { H ( r O ) . H ( q ) < i 1 ( x 2 ) . ...} is an orbi t of f.

Proof: Coiisider y* = H(xO) and {yo, gb g2, ...) the orbit of '/o under f. By induction.

Example 2 Consider the tent mnp

arld the logistic niiip g(:r) = -L:r(l - r ) (this firzctiorl is stctdied ln section 3.4). The

jtrriction H ( z ) = sin2(+) - is n conjugacg since ( g 0 H)( .r ) = s in2( r r ) = (Il f) ( r ) . From

Emrrqde 1. u le knnou thnt O a12d 213 are f k e d points of f a n d 21-5 nnd -I/5 c m period Iwo

pomts o f f : therefore H ( 0 ) = O und H(Z/3) = sin2(:) = ($1' = 3 -1 are 60th Jred points

o / g and H(?/.s) = sin2($) and H(-l/s) = sin2(%) are 60th period t u o points o j g .

In orcler to define t he concept of *-cfiaotic orbits". n-e introcluce a nurnber. the Lyu-

p t m o L! esponert t (or L yapuri u r nuniber) and t lie notion of serlsitivc dependence. Sensitive

dependence on initial conditions refers loosely to the property that pairs of points. ini-

tially close togethes. wil1 event.iially niove apart as ive iternte f.

Definition 6 Let f be a map on R. .4 point xo has sensitive dependence on initial

conditions if there ezists d > O ssirch that any neighborhood N of ro contains a point x

such that 1 f "x) - f "xo) 1 > d for* some nonnegative integer ?i.

Figure 3 Tinie Evoliition of .ra = O. 17 aricl go = 0.17 + 10-hnder g(x) = 4 r ( l - z).

To illustrate this property. i r e clepict in Figiire 3 the tinie evoliition of two nearby chta

iincler the niiip g ( r ) = -I.r(l - .r). Tlie differencc be tnwn the t ~ o initial points is 10-~:

i t could account for tlie ~iirious sources of irnprecision or error. \.ire observe tliiit after. a

certain time t lie tivo ciirws cleviate: so. the point r o = O. 17 has sensitive dependence on

init i d conditions.

In orcler to qiiaiitify siich a beharior ire iriti-ocliice the Lyapunos number lshich gives

a n average rate of separation of points r very close to some specific point .to.

Definition 7 Let j be n srnooth rnnp oj'the renl line R. The Lyapunov niimber L(zo)

of the orbit {ro. .q . ... } is defined as

L ( x O ) = lim ( l f ' (xo) 1 . ~ ~ 1 J ' ( : T ~ ) 1)lin. n - x

if this linlit exists. The Lyapunov exponent h ( x o ) is defined as

1 = n-w lim -(ln n 1 f '(.zo) 1 + ... + ln ljr(zn)

if this limit exists.

Sote that the Lyapunov esponent is an average dong an o

1

bit (a time average) of

the logarit hm of tlie derivat it-e. \ le v;iIl see i n the nest section that thjs average is equal

(under certain restrictions) to t,he space average (the integral) of the logarithm of the

derivat ive for alrnos t every point.

Remark: There esists an order 4 defined on the nat iiràl numbers W called the Sharko~vskii 's

o rd e ring:

This ordering on N. uhere * clenotes ordinary multiplication. allows the follo\i-ing result:

.-lssoriie f is a continiioos map on a n interval iind has n pcriod-p orbit. If p i y, ttien

f has it period-q orbit. .As a rorollar): we have that if j lias a peiiocl-3 orbit. then it

lias a period-n orbit for al1 integeis B (since 3 is tlie srnallest integer with respect to

Sliarkovskii's ordcring). For ii proof of tliis resiilt. sec [12. section 3.11.

2.3 Probabilistic Approach

In the previous section. ivc Iiaw introdiicecl the coriccpt of Lyapiinov esponent as a

particiilar propcrty of the orbit of n point .ro iinclei a fiincrion j. The follo~ing resiilt

tclls LIS tliat t lic Lyapiiriov esporicni cliiiracterizes tlie f~inction f i rself. t lie concept

becornes tliesefore niore iisefiil since it gives inforniation on the map indepenclently from

t Lie orbits.

In order to present ttiis remit. ive briefly introtluce the concepts of ergodicity and

inzarlant rnenszlre. -4 fiill description and proofs of the resiilts citn be foiind in [9. 11].The

iclea behind a probabilistic description of' a dynarnical system. f : I - 1. is to consider a

srnall region of points in I (insteacl of a single point) tliat a-e iterate iinder f. The central

qiiantity to evaliiate is the probability density p,(x) to be in state z at time n. Reciill tliat

a probability density is a normali:ed Lebesgue integrable function? (p,(x) l c k = 1. The

probability density p, (x) evolves in time and. according to the so-called Frobenius-Perron

equation [9]? we get: ..

wtiere S(x - / ( t ) ) clenotes the Dirac measure at x = f ( t ) .

\\'e cal1 the invariant probnbilit y density the espression p ( r ) t hat satisfies

sec [ I l ] . This clensity is particularly interesting since it indiices a measiire p ( x ) tliat is

preserved by f (by "preserved" we mean that p (C1) = p(C') uhere C" = { x : f(x) E C}).

This measiire is kno~vn as the ini;ar-iant niensure ancl is espressed as follows:

for C a Borel set.

.-\ dynaniical systerri is callecl ergodic if every invariant set C'. (f(Ct) = C ) . is eitlier of

nieastire O or 1. In ot her worcis. i f a e cannot clecompose the c l o m a i n of J into invariant

nieasiirable siibsets otliei. than the two previoiis orles.

if O 5 r < 112 Proposition 2.3.1 Thp terit rnnp /(.i.) = is erpodic.

if l l 2 ~ x ~ I

Proof: This is stiown in Chapter -1. Theorem -1.4.1 O

The follou-ing theorem establishes a relationship between ergodicity and p(x) .

Theorem 2.3.1 If the dynamical system / ( : r ) is ergodic: then there is exactly one in-

variant probability density p(x) which is Lebesg-ue integrable. Furthemore, if there is

u -unique inuariant probabdity density p(r) and p(s) > O almost euerywhere. then f is

ergodic.

ProoE See [9]. tl

Exarnple 4 Consider the tent map f (3). (see Ex 1). Then (2.1) becomes

uthich admi ts the solr~tions p(:s) = I( for a n y constant K . ICé nmst nornialize to get

the inrlor-ian t prohabilitg rlensity p ( r ) r I . i t is unique according to Proposition 2.3. i

nrid Theorenr 2.3.1. The intl(riant meas-trre of a s d s e t C for* the tent m u p is thus the

le,,& of the: cor-responding inter-clal (Lebesgue rneclsui-e). One can der& Jsorn this result

the incarlant n2eusur.e of the logistic map. Recul1 that the t e ~ t rnap generates the logis-

tic map thr-ough the transformation y = $ arcsin(&). (Exunple 2). h d e e d . using the

consenution probubility. [13/. we u7lte

This inruriclnt probability clensitg is uniquely deter-rnined and p ( r ) > O for al1 x E [O. 11.

Hence. by Theosrem 2.9.1 ute haue that the logistic equation is ergodic. Moreover. p(x)

indirces the f o h w i n g incanant mensure for C a Borel set:

Remark : By "almost everywhere". we mean that the condition is fulfilled for al1 points

escept for a siibset of rneasure zero, the measure to be iised being the invariant. rneasure.

SOK! let us make some connection between the Lyapunov esponent and the invariant

rneasiire for an ergodic dynamical system. The folloaing result is ver)* iisefiil since it

iillow one to cornplite the Lyiipunov esponent of a niap withoiit actually compiiting any

orbit.

Theorem 2.3.2 1f f : 1 - I hns an invariant measwe 11: and if f is ergodic urith respect

to p . then the Lyapunoc? ezponenr hixj is constani ulrnost e~i:rrywhri~t u r d

Proof: This cari lie provecl wi t h Birklioff ' s Ergoclic T heoreiii. [31.

The point to iinclerstiincl froni tliis clisciission is tliat if t he mnp preserws a reasori-

iible riieitsiire theri the Lyapiinov esponerit is coristaiit iilniost ewrywliere. The wliie

JP(.c) In I/'(.r)ldx is the spiire awrage of tlic fiinction In l f f ( r ) l ; t he value f2(p) is the

timc awriige of tlie fiinct ion Iri / f'(s) 1 dong t lie orbit of p. Tliiis. for nny iritegrable

fiinction. the time average dong alniost ail orbits is eqiial to the spiice average of tlie

fiinct ion. This indicates t liar i f f (s) is t lie rule of an ergoclic clynamical system. the orbit

ol a point p niiist be clcnsc for alriiost al1 poirits.

Chapter 3

Logistic Family

\ ié ~ i l l clefine the logistic lamily and stiidy sonie of its properties. Gvery concept and

sesult presentecl in tliis cliapter are already known ancl are part of al1 basic testbooks

[5. 12. 1-11. They concern the logistic niap ancl thex are to be gerieralized oves two sets

of polynomials in the nest cliapter. \\é will see that a function as simple as the logistic

tiiap conri give risc t o a wi.y ricti mat liemat ical t heory.

3.1 Definition of the Logistic Map

The logist ic family is defined as follows:

where O < X 5 4 and f., : [O. 11 + [O. ,\/JI.

As c m be noticed. O is a fised point of /,\ for al1 values of X and since [/'(O)( = I X I it

is stable for O < X < 1. In fact. every orbit of f,, is attractecl to the origin for O < X < 1:

Figure 4 (a).

Figure 4 (a) X = 112 (b) X = 2 (c) X = 3.5

The properties of f,\ change clrarriatically for rarioiis valiles of A. Consicler 1 < X < 3. ( A - 1 ) .I ( A - 1) In t liis case. sec Figure 4 b ) . t lie niap lias a sink at s = , since II,, (-\) 1 = j 1 - X I < 1.

(A - 1) For X > 3. as in Figure 4(c) . the fised point x = =\ f iç iinstable siiice 1 J,\(\(t) 1 =

11 - X I :, 1. The case X = -1 can be solïed in esact fasliion iinalyticiilly: ive will see it

latei in t liis chapter.

3.2 Bifurcation Diagram

The bijur-cdion diagrunt is ii coniputer image t hat captiires t lie ciynamics of Fi. a farnily of

functions. for many X-values. In that diagram. we plot the parameter X on the horizontal

asis versus the asyntptotic orbit of .ro under FI;', on the vertical asis aliem .ro can be an!.

point of the domain. By asymptotic orbit we mean that we plot the iterations of xo but

Ive do not plot the first few iterations (usually 100) of ro. That aa): ive allow the orbit

to "set tle down" and to reach its e ~ e n t iial beliûvior (k-periodic orbit, chaot ic orbit ....).

The dynarnics of the logistic farnily are ver!- comples as ive can see in Figure 5 .

Figure 5 Bifurcation Diagram of the Logist ic Fiinction

Fe\(.) = Xs(1 - x).

:\II t lie stable orbits are stiown in tliis picture. -4s esplairiecl in Section 3.1. O ancl y are L d i tisecl points for al1 valiles of ,\. 1-Io~ever. the tliagrani does not s t i o ~ the

part for O < X < 1 and t h e O part wtien 1 < A < 3 because the! are "unstable" fised

points for tliese specific values of A. Foi. 3 < X < 1 + fi. we see a per iod- t~~o sink tliat

also becornes unstable when X groin above L + J6 * 3.45.

I

r

1

I

I

1 , * 3

..' : 1

1

l *

4

- la l u 1s ll4 z *

Figure 6 F,,(x) = Xx(1 - x) for 3.8'2 < X < 3.85.

IG

The bifurcations give rise to period 2.48. 16, .. . (ptienonienon called period-doubling).

If ive zooni into the region 3.825 < X < 3-85 as in Figure 6. Ive see that a stable period

t hree orbit appears ancl ttien the period-doubling s t arts again to give orbi ts OF period

6 1 2 2 . This structure l i s the property of self-sirnil~n'ty, Le., some regions once

enlarged offer a resemblance to tlie entire graphic.

3.3 Lyapunov Exponents

The bifiircation diagram contains a large qiiantity of information for an entire Lmily of

fiinctioris. I t is as aell possible to ciiptiire in a single image the L>*apiiriov esponenrs of a

Earnily of fiirictions. \Vit11 the use of Tlieoreni 2.3.2. ripe c m compiite esplicitly the value

of the Lppiinov esporicrit of f,] for X = -1. Froni Esaniple -1. recall that the invariant

probability density of f.! is

For other X-valiies. ive neecl to find tlie invariant probability density whicli is not obvioiis.

Therefore, in orcfer to plot it. n e approximate the Lyapiinov esponent. To do so: we

cornpute the Lyapunov espanent of a fiinction f,\ starting with a ranclom value .r E [O, 11

for as many values of X as possible. Then \ire plot this value h( fa\) versus A. The starting

wliie x does not influence the result obtained (Tlieorem 3.3.2). In Figure 7. h( f,,) is

cornputed as follows:

where {zo, XI. xl. ...) is the orbit of 10 under f,!.

It is very interesthg to esamine the LJ-apiinov esponents of a farnily of fiinctions

versos the bifurcation diagram since t hey siniid t nneously express the behavior of the

clynnniical systeni. One can notice from Figure T t liat the Lyapunov esponent rises and

hits O at period doiiblings: at X = 3 and X s 3.45 for instance. It becornes positive in

the chaotic region (e.g.. we have h ( j 4 ) = ln(2) > 0) and it drops below O w hen t here is a

periodic attractor.

(a) Bifiircat ion Diagrani of t lie Logist ic Ftinct ion 1 . _ .

Figure 7 (b) Lyapiinov esponent of the Logistic Function versus the parameter A:

the Horizontal Direction is the Paranieter X between 2 and 4;

the Vertical Direction is the Lyapiinov esponent of fx.

3.4 X = 4, Exact Results

\\é ciin espress esplicitly ail orbits of the logistic ecluation in terms of simple f~inctions

ivlien X = 4, i.e., for f4(x) = 4 x ( l - x). Since x, E [O. 11. let r, = sin2(xen). Tlien

Tliat is ecpiivalent to O,,,, = ?O,, niocl 1 which lias the esplicit solution

0 , = ?"Oo mocl 1 (3.1)

arc5in where O. = +. It also $ives us a forrniila for the orbit of LI point :Q under Jl:

arcsin JT; .ru = sin2(2"do) where 00 = -

1 I

t f tw espress Ou as a binas- nuniber

" i do = 0.boblb2... = C ,. bi E {O. 1).

i=o - ttien applying (3.1) to O0 is equivalent to applying a left shift to O.bOblb 2 . . . ancl clropping

the integer part: ne cal1 t his fiinction. S. In other ~vords.

0, = -"Oo mocl 1

= 0.b,b,,lbn,2...

= Sn(O.boblh ...). wliere Sn = S o s"-~ for n > 1 ancl SL = S.

K i t h sucli an observation, it is fiiirly easy to find points of period k for any given integer k.

A rational number Ba E Q of period k generates an orbit of period k. i.e., XQ = sin2(.Bo)

is a periodic point of period k iinder f4 . In fact, since any rational niiniber is eventually

periodic. we have t hat for al1 do E Q {xo = s in2(do) x = f4 (so) , x- = f;?(xo), . ..} is an

event ~ially periodic orbit. The converse is also t,riie.

Example 5 Consider f4(4 = 4 r ( l - x) and xo = sin2(?). Ive have that Bo = f =

0.1101 101 10 ... in binaiy expansion. The initial condition Qo is a rational number. in

.4lthoirgh it is rzot o b c i o ~ frorn the ctalzre of Q = O.lSS255l. {ru, ri. .r.? ...} is a p e h d

threc orbit. BJ Theoren 2.2.1 we wncltrde that has chaotic orbits.

Theoreni 3.4.1 The logistic mnp Ji(.r) = -1.~(1 - .c) has chaotic orbits.

ProoE By Theorern 2.2.1 ancl Esample 5 . O

\\Wi the use of the binary expansion man- concepts become more int~iith-e: the

perioclicity of an orbit or the sensitive dependence on init i d conditions, for instance. M e

can show the latter as follows.

Consider O. = O.boblb ?.... l a = sin2(&) and d < 117 a positive real number. Mé

want to s h o ~ thnt for al1 5 > 0: there esists y0 such thet lxo - yol < E and Irr: - ykI =

I f . f ( ~ ) - 1 2 d for some nonnegative integer k.

Given E > O, u-e can find a 6 > O such that -

L by cont inui t~ of the function s i n2 (~x ) . Let k be the srnallest integer siich thût 3' > 5.

consider Be = ~ " o . b ~ b ~ b ~ . . . ) = 0.bibr;+ibk+2... and

< 6. Fiirtfiermore,

Chapter 4

Generalizat ion of the Logistic Family

III tliis chapter ive defirie two sets of fiinctioiis S and C. Wc s h o ~ that the dynaniical

beliavior of the niaps in S and in C is similar to the logistic niap's behavior: In fiict.

the logistic niap is a member of the set S. Ftirthermore. as for the logistic niap wlien

X = -1, t hese polyriomiiils have orbits t h c m bc sol\-ed esactly, t hus yielding analyt ic

closcd foi.miilas. Finally. aricl siirprisingiy. the iiivariant measure is the sanie for d l the

functions in S and in C.

4.1 Definition of two Sets of Functions: S and C

-4s we saw in Section 3.4. the logistic ecluation f4(r) = -lx(l - x) can be solved explicitly

nit ti the ilse of the change variable x = sin2(s;O). This cornes from t lie fact that

sin(20) = 2 s i n 0 J l - s i n ~ ~

sin2(20) = 4 sin2 8(l - sin' 0).

T h e same idea can be iised to construct anot her farnily ob tained by espressing cos2(20)

in terms of cos2 0: cos2(%) = 4 (cos2 O ) ? - 4(cos2 8) + 1 whicli induces the map g(z) =

41- - 4x + L and even the family gx = ,\(x2 - x + 114) : for X E (O. 41. More generally, it

is possible to build two large sets of fiinctions: one nhich expresses sin2(NB) in terms of

sin'% and the other one, cos2(!\'B) in terrns of cos". The first set, the Sine Fvnctions

Set is defined as fo1lon.s:

s = { S X ( r ) : S.hr(z) = s$(&) and !V = 1,2.3. ...) (4.1)

Remark: The eqiiations (-1.2) and (4.3) corne from the fact that

sin(n8) = 2 cos Osin((n - L )B ) - sin((n - - ) O )

cos(n0) = 2 cos O cos((n - 1)0) - cos((n - 2)O) .

This will be used later in this chapter. Xote that

s2 (~ ) = &(fi) = (?fifi)' = 4x(1 - x)

23

is the logistic map. hloreover, the polynornials C , ~ ( X ) , wit h the standardization ~ ~ ( 1 ) = 1.

are in fact the Tschebysheff Polynomials of the fist kind restricted to the interval [O, 11

insteed of [-1.11.

s,y(-c) =

if A' is even

uhem pn (s) are polynornials defined as

with pl (z) = 1 and p? (2) = 2 .

ProoE U7e show this result by incliiction on !Y. If N = 1. s&) = x = :cp&r) where

pi \,.ri = 1. I f .\* - 2. uc \ia\*e s,,.l.i = 2 . r n l / \ - .le' = . r \ e f l - . r 'p?( .~) i ~ h ~ i . ~ P?(x) = 2.

Consicter the resiilt truc for - 1 and 3- - 2 . Then. if N is ocld we have

where p H ( x ) = 2(1 - x2)piv-l (x) - jz) . On the ot lier hand. if A' is even. we have

Because of tlie square mots appearing in the definition of functions of S and of C!

rihat follo\\*s is an important aria l !.: ,ii-tri\.id resiih .

Proof: The proof for C'hT(z) is omit tecl since i t is ver- similnr to t lie proof for S N ( x ) .

Usinp tlie same notation as in the previous lemma. and letting Q S ( x ) = pIr(\n we c m

w i t e

- i x@i9 (x) ! if !Y is ocfcl S!V(.X) = ,si.(,,G) -

1 - ) ( ) il Ar is even

\\C can find the uliies of 9 ( r ) and h ( x ) esplicitly:

Since ai (s ) and 944 are botli polynornials. by (4.5) we conclude t hat cPip-(r) are ail

poly-tomials and so ore S A T ( ~ ) . by (-1.4). The degree of S!V(s) is !Y since the degree of

rvhicli is easy to show by indiict ion. O

Rernark: The construction of the polynomials to prove the previous result for C'AT(x)

is the following:

( ) if N is odd C!V(+) =

( ) if :\: is even

Kote tliat the alisoliite value of the leading coefficient. i.e.. the coefficient of the

liighest degree terni is -4"-' foi SAr(r) as w l l as for C'X(r). Hence. from eacti fiinction

S X ( r ) and C'iV(x) u.e define a famil>- of functions: for O < X 5 A"-' we have

S . : [O.Il - p.11 S A r (28)

"

The first few members of the sets S ancl C tire plotted in Figure 9 and are described

in the Table 1 below. Simply looking l i t the polynomials in the Table, one can daim

that S X ( x ) = CAr(x) when K is odd and. othernise, S N ( x ) and C N ( z ) add up to the

constant 1. This result and many ot her observations on the behavior of these polpomials

are proved in the nest sections.

Table 1 Few members of the sets S und C.

4.2 Particularities of S and C

The fiinctions in S and in C are constr~icted so that th- al1 have exact solutions. It

is. in fact, the principal idci: khind t tiis generalization. However: surprisingly. the sets

S and C eshibit interest ing algebriiic propert ies such as the closure under coniposit ion.

\Ve prove this in the present section, but first. we express SAT(x) and C!Y(z) in terms of

trigonometric hinctions in orcler to simplify the proofs.

Lemnia 4.2.1 Consider CN(s) and SN(z) as defincd in the pr-euiow section. CVe have

that

ProoE Sotice. by Trigonornetry.

sin(n0) = 2 cos O sin((n - 1)O) - sin((n - ? ) O ) (4.9)

cos(n0) = 2cosOcos((n - 1 ) O ) -cos((rz - ? ) O )

for al1 integers n 2 2. The result is obvious for = 1 and 2 . Suppose (-4.6) true for

.Y - 1 and :V - 2 . that is to s q ~ ~ - _ ~ ( s i n 0 ) ( ~ : ~ ) ~ [ i \ ' - - l ] (s in20)(y) - sin2((N - 1)0): and

the same eqiialities holcl for !Y - 2 . so

s,v-l (sin 0) = sin((!V - 1)O) and

sx_?(sin O ) = sin((AT - 210).

2 (2 \/1 - sin2 ~ s ~ ~ - ~ (sin O ) - sx&in O ) ) - -

Lemma 4.2.2 Consider- Cf:V(r) and SiV(r) as prer~ioasly defined, .we have that

S V ) = sin2(!\' iircsin J;) and

C:V jr) = cos2 ( !\i arccos fi)

for al1 r E [O. 11.

ProoE Let x E [O . 11. There esist O E [O. r /2] sucli thnt r = sin' 0. Tliiis. 0 =

arcsin fi. SO~L fi0111 Lemma 4.2.1 ive have t l i ~

The nest result enables ils to obtain a direct and simple relation between elements of

S and elements of C.

CX [ x ) . if A' is odd S!~(Z) =

1 - ( x ) if 1V is even

for al1 x E [O. 11.

ProoE Recalling from Trigononietry tliat arcsin 0 = w/%arccos O and tliat sin(&r/ii/'i

0) = k cos O1 we can w i t e thnt

= sin2(!\'i;/2 - A' arccos fi) cos2(-!V arccos JT), if !V is odd

= i sin2(-N arccos JT) . if N is even

\\é can clecluce niore froni Leniiiiii -L.Z.2 wliich allo~vs ils to classify the sets S iind

C. In fact. 1r.e have that S und C forni abelian rnonoids witli respect to the composition

of Fiinctions. o. Recall that a monoid is a non-enipty set :II together a-itli a binary

operation *. Le.. z r g E !II for al1 x. g E AI. Ttiere is an element e E :II. callecl the

identity element. for nhicli 3 t e = e * r = 2 for al1 .r E .II. Finally, the operation * is associative. Sloreover. if the operat ion is commiitat ive. we say t hat the nionoid is abelian.

In S and in C, the identity elements are SI($) = x and C1(x) = 2 respectively. The

composition of function is clearly associative. The Iüst conditions left to check are thnt

S üncl C are closed under composition and that this composition is commutative (which

is generally not the case with composition of functions).

Theorem 4.2.1 Consider ATi i\12 E M any positive integers. ?fi hare that

@en. take SNL (x) aiicl SlL(x) as defined previo~isly. Then.

Definition 9 Consider SX(:r ) E S and C'!Y(r) E C. ClTe say that S N ( x ) (C'N(z)

r.~spectivelg) is a prime eiement of S (ofC respectively) if !Y is n prime nunlber.

Corollary 4.2.1 .4ny polynoniial SXjr) E S nrid C'.Y(s) E C c m uniquely be erpressed

(1s a composition of prime elements of S and of C respectirelg. In this sense. {SiY(s) :

:Y prime) and {C:Y(.r) : ?Y prime) are called generators of S and of C respectirely.

Proof: Consider iV E N. From the Ftinclamental Theorem of Aritlimetic: we c m

iiniquely find prime numbers Pi, ...! P, such t hat N = PF' . Pfs. for O < ûi E N. Hence,

frorn the previous Theorem. ue clearly have that

Anet her useful observation that follon-s froni the Theorem 4.2.1 is t hat if one is inter-

ested in st.iidying the k-periodic points of a certain polynomial SAT(x) (or CN(x)). then

one only has to look at the fised points of the function S[iV"(x) since this 1 s t one is.

according t.o (Al?), s!v"x), (see Esample 1).

Remark: It is important to be ver! careful with the notation in order to avoid confusion:

SN1(x) could be interpreted as S [ X i ] (x) as well as ( S N ) i (2). For the rest of this poper,

the notation S N 2 ( x ) holds for ( S N ) '(x).

U C now construct closecl fornidas io espress x, = SlVn(x0), an orbit iinder S!V(x)

and y, = CNn(yo), an orbit ~indcr CN(x). xotice that formolas for orbits iinder prime

elenients of S and of C ~voiild be enough to deduce formulas under an). elernent of S and

of C becaiise of Corollary -I.2.1. However: we do riot proceed tliis way since it is simpler

to treat the general case (any integer :Y 2 0).

Theorem 4.2.2 Let {Q. XI. r?. ...} be the arbit of xo under S!V(x) and {go, y[: y?! ...) be

the orbit of go under C'n'(x). If we u r i t e 1, = sin2(;i0,) and y, = cos2(;i~,) we get that

ProoE The case N = 1 is obïioiis and the case A* = 2 is proved in Section 3.4. Mow.

to prove the general case, lkr > 2' ive Lise the Lenima 4.2.1:

This implies that Onil = N O , mod 1 for al1 n E N and al1 LY = 1,2,3, .... This eq~iation

has the soiution

For {q. 12. ...} an orbit of S!\:(z) and {yo. yl. y?: ...} an orbit of Cilr(x)! the last

t heorern enables ils to mite

arcsin J2o r, = s i n ' ( h ' " ~ 0 ~ ) where Bo =

e

for al1 n E N.

In the case !V > 1. i f !ire esprcss Uo in base iY.

" b, O0=C5 where bi E {O. 1. .... !Y - 1}

r =O

tlien (4.13) is eqiiifiilent to applying S (definecl in Section 9.4 : Le.. shift and tlrop the

integer part). It implies t hat if Bo = O.bobLb -... (in base iV) then s, = sin2(&) wliere

0, = 0.b,b,,lb,,2.... As for the logistic map. this notation allows us to iinderstancl the

nature of sensitive clepenclence on initial conditions. If we choose two numbers Bo =

O.bOblb ?... and $ = O.bOblb ?... very close to one another but different, there miist exist a

k E N siich that bk # &. IF the smallest sucli k is arbitrarily large then the difference

betu-een O. and & is negligible but it becomes more and more important as ive iterate Bo

and since bk and & are shifted to the left et each iteration.

4.3 Conjugate Nlaps of functions in S and C

As we esplaineci earlier in Section 2.2. a conjugate rnap contains a great deal of infornia-

tion. To be precise, it c m be used to transform a dynamical systern whose betiavior is

unknown or unproved into another one whose dynamics is known. In the prerpious chap-

ter, miiny results concerning the logistic map were easy to prove because of its conjugacy

~ ~ i t h a sirnpler rriap: the tent map. The same idea is iised for SX(xj and C!Y(r). This

is t lie major motivation of tliis Section.

Let T : Y ( x ) be ii piecewise lineiir fiinction defined on subintervals [ k / N . (k + L)/!\'] of

!O. 11 Iiy set t ing

\ M i k = 0: 1, .... S - 1: (see Figure 9). Sote tliiit, TZ(:r) is the tent niap.

Tlieorem 4.3.1 Consider S.\-(r) and C X ( x ) as defined in Section 4.1 und T.\'(x) de-

fined prwioi~sly. Clé haire that SA'(.c) and C'S(.r) w e both conjugate to T S J x ) . Le.,

t i w t erist JIs ( r ) tind H&) sitch that

2 C':l'a H, = H, O T N for M = 1.2,3 ....

uhe7.e H J x ) = sin2(?) - and &(z) = cos2(?).

Proof: First. note from Figure 8 that H,(x) = sin2(?) and HC(z) = cos2(%) are

both continuous and bijective maps on the interval [O, 11.

Figure 8 Conjugacies between S N ( z ) and T N ( x )

and between C N ( r ) ancl T N ( x ) respecti~el':

!\ i ï~i~;t ' 2 = ( i sin (?)) -

The resiilt for C'N(r) can be provecl sirnilarly. 0

Corollary 4.3.1 For al1 integeri; !V > 1. ute ha-ce that S N ( r ) is conjugate to C N ( z )

with conjugacy map H ( x ) = 1 - x.

Proof: This is a consequence from the fact that conjiigacy is an equivalence relation

but Ive show it esplici t ly. Kote t hat H (x) = 1 - x = HM' (x) = sin2(ûrccos Jr) =

cos2(arcsin m. Therefore. we have that

by Lernnia -1.2.1 and Lemma -1.2.2. Cl

In Figiire 9. below. wc tlcpict the praphs of C X . TAr ancl SN for 2 < A' 5 7. In each

case note t hat S N ( x ) is a ..srnoot ti" version of Ti\- jx).

? Figure 9 CN's , TWs and SS's for N = 2 , ..., , .

4.4 Lyapunov Exponents of S N ( x ) and C N ( x )

In t liis section we ciilculate the Lyapunov esponents of the functions S N ( x ) iind CX(z ) .

\ t é compare the plots of the Lyapuno~ esponents with the bifurcation diagrams of the

families of fiirict ions SNl (x) and C:\i (x) . Mowever, ive must previously find the inmriant

measwe of those fiinctions.

Theoreni 4.4.1 Consider f : [O, 11 -+ [O. 1). Let ( I j ) be a finite famil9 of disjoint open

i n t e r - c a l in [O. 11 such that

4. Ther-e exists 3 > 1 and no > O szrch that if x,, E U,I,. (:cm = /'"(xO)). for d l

O 5 m 5 rio - I uqe hace (( f "O)'(x0 j 1 2 3.

5. There ends rn > O s-uch that .A4 (f - m ( ~ j ) n li) # O for e t w y i. j .

6. There erist C > O and O < 7 < 1 s-uch that

Shen ther~e exists a unigae f -inruriant probability measure p ,tuhich is absolutely contin-

uozrs wzth respect to Lebesgzle measure. This also irnplies that f is ergodic (accordkg to

Theorern 2.3.1).

ProoE See [IO]. Cl

To find the invariant nieiisure of S'!Vix) and of CN(z)! Ive proceed as we did in

Esaniple 4 for the logist,ic eqiiatiori. i.e., u-e first show that the maps T - I ( x ) are ergodic.

Corollary 4.4.1 The tent rnaps TAr(x) are ergodic, for Ar 2 2 .

Proof: Consicler T N ( x ) n i t h !V 3 '2. We have that TRr(z) satisfies the conditions 1

to 6 of the previoiis Thcorem. .\Lm!. conditions are trivial, we show conclitions -1 and 5

ordy.

For T X ( r ) . 1, = (+, T) . j = 0. 1. . . . . A r - 1. Hence. uJ, = [O' l]\{j/X : j =

0. 1. .... h'). Take 3 E (1 .2) . K c liaw tliat I/'(.r)I = !Y. For al1 z. E u,l, so

To prove coriditiori 5 . takc rn = 1. We ha\-e that

Xow. let ils find t lie invariant probiibility clerisity of the fiinction TAF(x):

I f !V is even,

n-hile. if N is ocld.

for ahicli p(z) = 1 is the unique normalized solution. since TN(r ) is ergodic (see The-

orem 2.3.1). The invariant measure of T N ( x ) is thus the Lebesgue measiire. Kow!

because &(s) = sin2(Y) - is the conjugacy map of T N ( x ) and SiV(x), ive have that (see

Section 2.3):

but witli p(y ) = 1.

which indiices the invariant nieasiire

wlieie Cf is a Borel set. see [13. 2.11. Since H C ( x ) = c o s 2 ( 7 ) is the conjiigacy of T!V(r)

and CtA'(.c) WC have that

PP) = T & I - x)'

which induces the invariant measure

ii-liere C' is a Borel set. Hence ail the f~rnctions from both sets S and C ha .ve the same

invariant measure that we denote p = p s = pc.

The result we want to show at this point. concerns the Lyapiinov esponent of SX(s)

and of CAT(r) which is calciilatecl to be In(:\:). To do so, ive need to prove few lemmas.

but first. we conipiite the Lyapiinov esponent of S N ( x ) .

Let us define = { k / X : 1; = 0: 1, ...' N}. This is the set of points for which TAr(r)

is riot differentiable and for n*liicli SS'iir) iiiitl C W ( . T ) are zero. Xote that the image of

a point x E VLv iinder S N ( z ) or C.'X(x) is either O or 1.

Lemma 4.4.1 For g i r e n :Y let (xo. :r l . r-. ...) be an orbit such that ri $ DLv f'or i =

0. 1. '1: .... Tlien the Lyapunou e q o n e n t o j k O is 1rijh7) under T N ( r ) .

ProoE \\é have ttiat (TX1(.r:jl = / k XI = :Y for al1 :r E [O. l]\D';. Le.. for al1 r foi

aliich tlie tleriviitiw of T.\*(.r) is tlcfiiiecl. Consicler ro sucti tliat the orbi t (xo - C I . 1 2 . ...}

iincler T.\:(.c) is siicti t hat r, 'PY for i = 0. 1. '2. .. . . Tlien

hi orclci. to generalize this liut Leninia to t h e fiinctions of S and of C. 1ve neecl to . ,/5 clefine two more sets of points. Let = { r E [O. 11 : E Q } and Vc = {x E [O. 11 :

arccos /Ï - E Q} be these two sets of points. The rest of the argument is done for SX(xj

only sinre tlie sanie reasoning enables iis to prove the resiilt for CA'(x).

Lemma 4.4.2 The set Vs h.as Lebesgîle measwe zero.

ProoE The function g(x) = JrCsin - bG being bijective on the interval [O. 11. ive have

t hat tlie Lebesgue rneasure of Ds is equal to tlie Lebesgiie nieasure of the set g(Ds) =

{ g r : x V } But. by definition of Vs, ne ha\+e that g(Vs) c Q n[O. 11 which have

Lebesgiie m e s u r e zero. E

Lemma 4.4.3 Let xo E [O! 11,. xi = S:\"(xo) and N > 1. Ifs $ Ds then the orbit of xo

imder SN(r ) is not asyrnptotically periodic and xi O or 1 for al1 i = 0: 1.2. ....

Proof: Froni Theoreni -1.2.2, recall that if ive write X i = sin2(n0,). then Bi =

iViB0 rnod 1 u-liere do = e. By continuity of t lie fiinction e. u-e Iiave t hat the

orbit {I,} is asymptotically periodic if and only if the orbit {Bi} is. Kote that { B i } is the

osbir of do under t lie Euiict ioii j ( : c ) = X.r moi1 1. 1-Icnce, according to Lcmrn~i 2.1.1 y since

1 (j")'(.r)l = :Yk > 1. \dieri tlic tleri~atiw eesists: x e hin-e that if ( O , } is nsyniptotically

pcriodic tlien it m~ist be event iiitlly perioclic. But. clearly. (Oi} is e\*entiially periodic if

ancl only if Ou E Q so if :CO E Vs.

Fiirt lierniore. note t hat .ri = O or 1 i f and only if Qi = 0. 112 or 1. So slippose :r,, = O

os 1. n e have V i 0 = 0. 112 or 1. But. this iniplies tliiit tlic orbit { O i } is cventriiiily periodic.

so Q0 E Q aiicl .ru E 'Ds. 17

Remark: Tlic clefinitioti of the Lyapiinov esponent of a ftinction /(I) iit .ro is

e e , = f ( j . Using the Chain Ride rhat tells LIS that ( j ' ) ' ( ro) = nYzo / l ( x , ) we

can svri te tliat 1

h ( ~ ) = Iirn - In 1 (f ' ) ' ( q ) 1. n--x 11

Theorem 4.4.2 The Lyapunou exponent of S X ( x ) and CX(.r) is ln(Ar) for almost al1

x E (O. 11.

Proof: As iisual. Lve restrict oiir proof to S!Y(x). the proof for C'iV(s) being similar.

We prove that the result is triie for al1 x $ a. ive airead). p ro~ed that the set Vs

has m e s u r e zero. Take xo E [O. I]\Vs. Recall that SAr(x) = Hs 0 TAT o H;L(2) and

S N i ( x ) = H , ~ T I V ' ~ H ; ~ ( X ) wliere H s ( z ) = sin2(?). - Kote that IHs(x)l 5 1 for x E [O. l]

(sec Figure 8). We use the notation yi = H;'(x,) where xi = SNi(xo). \\é have that

go $ Q since ro $ Ds and so? T!\"(yi) = Ar for al1 i = 0, 1.2. .... Hence.

1 h(xo) = lirn - ln I (SArn) ' (zo) 1

n - x n 1

= lim - ln I(Hs O S N n o I < ; ~ ) ' ( Q ) I n-ca n

On the otlier liaricl. since .co $ Ds ancl according to Lemma 4.-4.3. we ciin choose a

secpierice of integers ri, going to infiriity siich tliat .cnJ E [5. 1 - 51. .-\lso lH;(.cj 1 > O in the

open interval (0. 1). so for al1 5 > O tliere is a boiiiicl lij > O sucli tliat Ii',j < IHi(r) 1 for

.r E [J. 1 - 51. Sotc tliat if r, E [d. 1 - SI. tlicri y i = fI;'(:r,) E [S. I - CS] since f13:1(r) > r

\dien .r E (O. 112) ancl H $ : L ( ~ j < .r phen r E (1/2. 1) . Then.

1 = n,- .~ lim -(ln~~~~(yr,,)~+ln~(T~~~~~)'(go)~+ln~(H~~1)'(~o)~) n -

I

1 2 lirn -(ln(K,j) + nj h ( N ) + ln 1 ( H s ' ) ' ( x O ) 1)

" J - ' ~ 'n j

Mence! h ( r ) = ln(iV) for al1 z $ % 0

Rernark: The last theorem is an important fact for oiir stiicly. Indeed, it confirms

that the fiinctions SN(x ) and C N ( r ) are not al1 conjugate to each other. Furthermore,

considerinp the f~inction

tliat S!\r(rj (or C'!b7(x)) to i ts Lyapunov esponent, and tising the Theorem - E l ,

we have that

Tliat is to say, Ii is an Iiorr~omorphisrri of rnorwids froni the monoid S (or C). nit li respect

to conposition. to the monoid R. witli respect to addition. In f x t . it is an injective

honioriiorp tiisrn wliich enables ils to say t liat t ticre is an isoniorphisni of iibclian moiioicls

froni S or C to the image of h in 1. wliicli is {ln(!\-) : A- = L! 2. :3. ...}.

Lyapunov

Esponcnts

of

S., (.r)

Bifiircat ion

D iiigram

ancl

Lyapiino\-

Esponents

of

S3.\ ( 3 )

ancl of

C3A (4

Bifurcation

Diagarn

ancl , 1. . 8 , 1, , 16 .

Lyapiriov

Esponents

of

CU,, (x)

â ion Bifurc t '

Diagram

ancl

Lyapunov

Esponents

of

s4\ (4

Figure 10 Bifurcation Diagrams vs Lyapunov Exponents

Chapter 5

Conclusion

\\é have foiincl two abeliiiri monoids S uncl C of non-conjiigate polynoniial niaps iliat

eslii bit diiiotic beliavior : the). have chaotic orbits (escept for S 1 (x) ancl Cl (x)). Indeed.

a chaotic orbit is one ttiat is noi. iisj.niptotically perioclic ancl for aliicli the Lyapunov

esponent is greater tliun zero j 141. The Lyapiinov esponent of S:\'(r) and C:Y(x) is

hi(:\') (Theorcm -1.-1.2). .\loreover. clioosing sa = s i n ' ( ~ 0 ~ ) (or go = cos2(~wo) ) with

OU $ Q (or $ Q) assi~res tliilt the orbit of . r ~ [(or go) is not iisyiiptotically perioclic

(: Leriinia -1.-1.3).

.-\notlier way to convince oiirselves that the polynomials of S and C have cliaotic

orbits. is to fincl a number ro of period 3 for each polynoniinl in the sets (Theorern 2.2.1).

For a fisccl 1 < !V E N. consider the ntimber

= 0.001001 ... (:in base N).

Hence. t lie orbit of xo ~inder SiY(x) is z, = s i n ' ' ( : ~ ~ a ~ ~ ) , that is

Tlie point xo generates tlie tlesiretl periocl 3 orbit. We thiis showed ttiat the rhaoticity

pr0pei.t ies. originally lound in t lie logis t ic niapl c m be estendecl to infini te families of

polynoniiiil niaps. naiiicly S ancl C. whicli include the logistic map.

Riit licrriiore. t hroiigti t liis t licsis. we tiaw mtrictecl oiir st iidy to fii~ict ions of the

form S!Y(.r) = s in2(X arcsin fi) and CP:V(:r) = cos2(!\' arccos fi) for LI' a positive in-

iegcr. OIE coiild Lic intesestecl in generalizing tliis to an)' fiinction of the type Sû(.c) =

sin2(a LI-csiii \6) and Cn (s) = cos2(n arrcos fi) for o E Q or even ci E R. However.

as for t lie fiinct ions Sl(.c) aricl C'l ( r ) . t lie fiinct ions Sn(.r) or C'a(.rj clon't eshibit an!.

iriteresting propertics for O < a 5 1.

For instarice. the fiinct ion S!(.T) = sin'! l!.) arrsin &j is. in fact. conjiigare to the

fiinction g(,.r) = 4 2 . \vliicIi Ilas no fised point (other than zero) and no perioclic point:

al1 the orbits uncler S; - jx) are attracteci to the origin. Hence. Sh(.r) - doesn't have an'

chaot ic orbits.

To generalize the resul ts of t his t hesis to fiinct ions of the form Sa(x) and C h ( x ) . one

coiild try to ansiver the follo\~irig qiies t ions:

1. 1s In(a) tlie Lyspiinov esponent of Sa(x) and Cc+) for every n E R'?

2. Do Sci(x) and C'a(r) admit chaotic behavior if and only if In1 > l?

I t is not possible to answer this last question iising esactly the same approacli as for

SN(x) and C K ( x ) for N > O. an integer (see Tlieorern 4.2.2). Indeed? t,o prove the

cliiiotic beliavior of the functions SN(s) and CN(z) we used the espression of a point

in "base K" nhich makes sense if A' is an integer greater t han 1 only.

The families {Sa (x ) : a E R } and ( C Û ( L ) : a E IR} might aell generalizecl the idea

of the present thesis.

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