AD-A261 427

C

NPS-MA-93-006

NAVAL POSTGRADUATE SCHOOLMonterey, California

7T"C R AD%3

0BOUNDEDNESS AND ASYMPTOTICS M

OF THE GENERALIZED LO_THEODORUS ITERATION

by

Jeffery J. Leader

Technical Report For Period

March 1992 - June 1992

Approved for public release; distribution unlimited

Prepared for: Naval Postgraduate School943 I4 • Monterey, CA 93943

148 i II II l IIII I

NAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943

Rear Admiral R. W. West, Jr. Harrison ShullSuperintendent Provost

This report was prepared in conjunction with research conductedfor the Naval Postgraduate School and funded by the NavalPostgraduate School.

Reproduction of all or part of this report is authorized.

This report was prepared by:

(4-"C,JOF~yfA,- r/EADER/,ýProfeiss6rz

Reviewed by: Released by:

;.IC4HARD ZFRANKE PAUL J. MARTdChairman Dean of ResearchDepartment of Mathematics

DISCLAIMKI NOTICK

THIS DOCUMENT IS BEST

QUALITY AVAILABLE. THE COPY

FURNISHED TO DTIC CONTAINED

A SIGNIFICANT NUMBER OF

PAGES WHICH DO NOT

REPRODUCE LEGIBLY.

Unclassified

SECURITY CLASSFCATON Of 7H!S PAGE

REPORT DOCUMENTATION PAGE O No 0704.0188

la REPORT SECURITY CLASSF'CATION 10 REST C',VE MAP.,.G,

Unclassified2a SECURITY CLASSIFICATION AUTH.ORt7Y 3 DiSTRi~i-iQ A,-A A-6 Vy Q: IPO'G

2b DECLASSIFICATION, DOWNGRADING SCHEDjLE Approved for public reliease;distribution unlimited

4 PERFORMING ORGANIZATION REPORT NuMBER(S) 5 MONITORN•G ORGQA,'A2•,G% REPQ ' "'SNPS-MA-93-006 NPS-MA-93-006

6a NAME OF PERFORMING ORGANIZATION 60 OFF CE SYMBOL 7a NAME OF M NC',R ©DRGA; ZIa' "N

(if applicable)

Naval Postgraduate School MA Naval Postgraduate School6C ADDRESS 'City, State, ard ZIP Code) 70 ADDRESS (City, State and ZIPCooeý'

11onterey, CA 93943 Monterey, CA 93943

8a NAME OF FUNDING ! SPONSORING Bb OFFICE SYMBOL 9 PROCQREMENT I NSuRLM•7.D ,-%4 CA',O' .,VaER"ORGANIZATION (If applicable)

Naval Postgraduate School MA 0&&N Direct Funding

Bc. ADDRESS (City, State, and ZIP Code) 70 SOi.RC" nc K 2 ".G P"?%' . ..Monterey, CA 93943 PROGRAM -';OzECT 0,s• E' CELEMENT NO INO jNO ACC•SSG% NO

11 TITLE (include Security Classification)

Boundedness and Asymptotics of the Generalized Theodorus Iteration

12 PERSONAL AUJTHOR(S)

Jeffery J. Leader13a TYPE OF REPORT l3b TIME COVERED 4 DATE OF REPOR7 Iy'ear MOnru Day) P GE CC-%-Technical Report FROM 3-92 TO 6-92 15 Dec 92 20

16 SUPPLEMENTARY NOTATION

17 COSAT• CODCS '8 SUB;EC" TERMS ýContinue on reverse if necessary ard idenrrt by biock number)FIELD GROUP I SIJ-GROup Chaotic iteration, Strange attractor

19 ABSTRACT (Continue on reverse if necessary and idientify by block number)A weakly chaotic iteration, called the Generalized Theodorus Iteration, is

analyzed with respect to its boundedness and asvmptotics. The limit sets, whichoften are strange attractors, are also considered. Applications are discussed.

* 20 DISTRIBUTION rAVALABI',ITY OF ABSTRACT 2 ABSTRACT SEC'JITY C ASSiP CATONr3UNCLASSIFIED,UNLIVITED 0 SAVE AS RPT [] DTIC US;QS Unclassified

22a NAM•E OF RESPON.S-BLE" %NDIVO UtAL 22b TELEPHONE (include Area Coa,7ie')' : 02c O C . SYvso 'LJeffery J. Leader (408)656-2335 1MA/Le

OD Form 1473, JUN 86 Previous editions are obsolete SECUe-' CLASS ; C(A 0 r :-.

S/N 0102-LF-014--6603

arii Ayr!pto~trof the Gener-al 4. Zd T~~k

Iterat- on

* JEVF-FERY J. LEADER

D~partment of Mathematics, Naval Postgraduate -School

Abstract: A wt-ka7y chaotic iteration, called the Generalize-d

Thoodorus Itepration, is anal yzipd with respect to0 it's

boundedness and asymnptotics. The limit sets, which oiften. are

strangea attractors, are also considered. Applications are

di SCUSv-ý 1-4.

Key words: Ch-aztjc jteatIjOr, - raig at a '

NTIS CRA&I

UJflailh1OIr~te

Otisribu~turl i

Av~iabiflityt Codes

I , •NTRODUCTION

In [2)., Plato states that his teacher, Theodorus of Cyrene,

was th., first individual to prove that V3,-c,.. -V17 were

irratic:--al IL. p.•J. Since then many scholars have wondered

how he did this with the mathematical tools he had at hand,

and, pFrhaps more puzzling, why he stopped at the number

seventeen. An answer which is almost certairly incorrect but

which is nonetheless intriguing was given by J. H. Anderhub in

1918 (see [3.5,e,10]}. He suggested that Theodorus might have

used the following construction. Form the triangle with

vertice.: at (0,0), CI,0), C1,1). Note that the hypotenuse has

length hiY. Starting now at the point (I,1), draw a line

segment of unit length perpendicular to the hypotenuse and in

the counterclockwise direction; this gives a new triangle with

this line as one side, the previous hypotenuse as the other

side, and a hypotenuse of length -43. Continui ng in this way,

a spiral figUre Ccailed the q11drt37rze1schrueci~e rio;, or

square-root -snail) emerges with the numbers In as the lengt hs

of the radii (Fic. 1). The last hypotenuse drawn before the

figure overlaps itself has length -V17, and this, it was

suggestced, was why Theodorus stopped at seventeen Csee [5D3.

Although this is almost certainly the wrong answer, the

sequence of points formed by this process was studied on its

own merits by Hlawka C(10]; see also [E24). He proved a

number cf results concerning this construction. The study was

then pirKed up by Davis [3J, who noted that the points could

i Z, tZ

* . . • .k ...~.. • ~........ ¾.. • ... Ja : th. ' • •.F..

z : = z (1 C.1)

Cwith 0 :1) and that they we; i asymptotically an Archimedean0

spiral. He then interpolated an analytic curve to these

points, which he called the Spiral of Theodorus.

Noting that (1.1) could be considered Euler's method for the

ordinary differf-'ntial equation

z = zitzl

with urI- st•.e., size. .VIT sugCest ed a• a general zata or. of

C.I t. e iter-ation

z = c:z f~z /]zIZr,. ,t = xr-, + ,nr•~

with o,(•C and z a giver- conple number. This iteration,

called the Complex Generalized Theodorus Iteration CCGT-ID, is

studied in [1,15,20]. Davis later suggested the further

gerneraii •zataori

V - A•'' + B'V /','i'" I, C1. *:

where P -.d B are real mxm mat.rice., V ji a civen nonzero

m-vect cr , and 1i Ii i S the Euci idearn vector norm. Thi s

itera-ti -n, name7H the Ge-rneralized Th.ydorus It.erat: •r, Ci:

the present author, displays a great variety of strange

attractors for appropriate choices of A ar,: B (Figs. 2-•. and

include.- the complex GTI as a special case; it is studied in

[4,15,1-3,19). In this report will summarize some of the

informa .ion contained in [18] and L19I Cadding some material

from C,51) with respect to sufficient boundedness conditions

and as.mptotic estimates for this iteration. Since w-- are

intereE ed in boundedness and asymptotics, in stating our

result! we will ignore the case where V is such that. V i!0

the nu' I vector for some n (in which case the iteration

stops}

2S. BOULNDEDNESS

Let us try to find sufficient conditions for boundedness of

the GT-I. Taking norms on both sides of C1.2) gives

11V 11 < IIAaI. -IV 11 + 119 P

Iterati-ng this relation, we find thatfl-I

INV II < iiAI' A - IV Ii + 1B11.E 11A11' (C2. 1)%. 0

and clearly if IIA11<1 then in the limit the norm is bounded by

IlB11/(1 - HAII)

Hencre iAll<i is sufficient. for boundedness of the orbit, (for

any V . From this and the similarity of (1.27) to the linear0

F yst. err

V = AXV

which t as a bounded sol uti on for every V when the spectral

radi us of A is lest than unity Cand at least one unbounded

solutic -, when the spectral radius of A is greater than unity),

it seer3 reasonable to conjecture that p(A)<( suffices for the

boundedness of (1.2D), where pCA) is the spectral radius of A.

That this is in fact the case follows from two lemmas. We

omit th-? rather technical proof of Lemma 1. It is not hard to

establi:h this result by using an approach similar to that

",,., J.. f:e andslxb•,b-u• it is rather tedious to

carry cit. the proof in detail (see [151). Lemma L is a

discret, versicom, cf an ordinary differential equations result

4

LEMMA I If 1;V 'i is unbounded then liV ;--:.

ELMMA a L.4.t A be a matrix with p(A)<l and suppose that B~r,)

is a sequence of matrices with 1IBCn) n--O as n--*. Thar. al1

sol uti o: s of

z = CA - BCnD- zr,*4

tend to zero as n--x.

PROOF: Since pCA7l'. there exists a matrix norm II such thatA

IIAR <1. We will choose this to be a natural matrix norm andA

wil l a] so denote by 0 • b the veztor- norm which induce it.A

Let e equal CI-IiAI A. Now sircc all matria X norror- areA

eouivalent. inr the-- sense- that if 6147n'11--- then li H .n l--O, we

have from our hypothesis that ViPCn:)1 -AC,. Hence w; can chooseA

•r-, N" suc:h that I!B<;C "D B -'7" for all t ..

o'7,' c-.nider itcratino

z = CA + P.CF, -•.h

Cfor sore z C N times to produce thc vector z . t .=z ancý0 N 0N

C= A + BC N + nDi 'w,

V C•O. I aearly t.,o =z for all ??O. Now

,, CA + PCN + ng)-) -*-A + B1N ,)*D,'

and so

1 II A + BCN + nr)I 1 • iA + KCN)II 1• o 11r*1 A A A 0 A

= li- 0 II A n A + BCN + i)lio A SA

,- =0

Tr~en

ti'! < l•1A f ( IABI + IBIBCN + i)lB ):S It 0 A A A

0:

= l A, il r Cl - =":T

.=

w)-i ch ,-!ear- y tends to zero as r--s. Henze lI1 1! -- 0 as rn--*ar A

sr. thai. liz 0' --- 0 as n--tm and so by the equivalence of vector

norms on [-K we have that 11z 4--,0 as n--. This completes the

proof. o

We may nos.' -sh_,., the desired result.

THEOREm 1: If p(A)D< therm the solution of C1.2) is bounded.

PROOF: Suppose the contrary and choose a V such that 11V II is0

unbound-ed. By Lemma I , l1V 11-OM. Define BCnr=BzIIV i. By the

hypoth.ss, IIP1-r,T) i b--:. Consider the iteration

z = CA * BCn))*z

with z (=V . Clearly z•=V for all n. The conditions of Lemma

R are met and so liV II--%0 as nr--* . But this contradicts IV 1i

unbounded, which establishes the result. o

It. i_ also possible1, to show that if pCA')>1 then there exists

a V s-.;c t-.- i1V 1i is unbounded Csee scection 5> . 7n thi.•0

sense he 0-5 is surprisinlyv similar to the Ii near system

c&. u'..

In t..- - part.icular case A=C a de.tailed analysis is possible

E- 1 7 Th.s case i- relat~ed to the power met.hod for the

numerical sollition of eigpnvalue problems.

3. AsYMPTOTTCS

This leaves the question of boundedness for the case pA)=1.

It is possible to construct examples showing that when A is

unispec-,ral the orbits of the C5TI may be always bounded for

or,~ e¢ho.ce of B and always unbounded for another choice of P.

Howev-r, a gerl st.;t.emnTt about the a-ymptot jc'.- of the

p(A)_i and any eigenvalues of A that lie on the unit circle

are simple Ci. e. have as many linearly independent

eigenveztors as their multiplicity). In this case the linear

system C2.2:) has the property that all solutions are bounded

(althouigh in general the bound will depend on V ). Before0

stating the corresponding result f--, +he GTI, eze state:

HuIuWARA-s THrORrM: Let- z be any solution of

ttz = (A 4 ~)~ 3•

for t=I,1,, .2.. where A: - s a constant matrix and

F liBt)U K< on

Supposc- that. A is of bounded type, so that all sclutions of

yt=A*yt_1 are bounded for t>0. Then every solution of (3.1)

is bounded.

T11 s resul t-, an analogue of a theorem from differential

equations;, may be found ri [i]. The proof there uses thk *I

vector norm, but the argument i! valid for any vector norm and

compat.i ble matrix norm. We will now use- it to prove an

as.ympt r .iv resuit for *.he GTI.

THEOIrM EC: Let A be of bounded type in C1.27-•-. Then

PROoQ: Suppose the cont-a.ry, i.e. that for some V we have00

1, lt1lV II < OD C3 2)

and construct such a sequence <V > by using this V and Ci.2).

Define BCn)=B/IIV iI and consider the iteration

Y CA + fonn-)tY

Where ;r- si:t Y =V . Cl earlIy Y =V for all rJ the two

7

sequences are identical. Now

E tB~n) E = P IiBI1IV 1I

r'=OO

r,=r,

= ,0

which by supposition is bounded. Therefore by Hukuwara's

Theorer,, Y I! is bounded, Then 1IV I! is bounded, that is, 3M>O

such that IIV 11<M for all n. Put then 1ll/V Ii>1ZM and since

i/lly II---O we cannot.. h ave C3i. s. This contradiction

establishes the result. D

ii fazt. a slightly more general result. holds. If l.11I is any

vector norm and I'C) is any function that. is continuous on

rO,OD and positive on (O,0D, and if A is of bounded type, then

any sol ut.io of

V = A*V + B*V .'1C 1IIV I)

must. be such that.

S1l/•CIiV lIi --nmor,=O

ThIs mara, be showr in a completely anal oou-o mnner c se, "15)

for det-.iis and applications..

]Toorem E? establishes that, when p1AT.=, but A i of bounded

type, if the orbits diverge then they dn so in a way that is

not. muotvi worse than linear (since the sum of the reciprocals

of the noHm -ri verges). In fact if IIAII=I in the spectral norr

then it. is a fact that the divergence is at most linear, as

can be seen by setting IIAII=I in (2.1) to get

lv II :S 1lV It + nliBl1S0

a•: the ;ymptot-ic estimate. We will later consider this point

in grea er deta.il when we show in section 5 tha;t if A is of

bounded type then the divergence is a]ways at most linear.

r, de to relf irne our a symptot ;c esti r, ', cs o~f t he prev;cous

sectio-, wie: w;.±i ner-d a formula for the snititior-, of the

a torat~ on- C1 .Z' Gi ven an iteQration of thQ form

z = Aýt)*z + (,

wi -h a a: d z vFe-tors and A~tD a timc--varving, matrix,

t 0

I. 9 S t+1 0

wherea c - th;--r ta condi tior, and fCY > is a furadamernt-al.

rn~xsept.. of S-oltr2onx¶; for

-zt=Aý,t zC 4. i

C tha t :S. y =1 an d y =ACr,-)*Y D [ai. For t he GTT , the

m,4t~rix A is cont-r 7, 01 =Vz Is gi ven, Fur*thermore, YA

t t

so that

v = An 1 -- I + _ A t'*E*V t, iV tIi 4.2

t = (-,

is the so7l ut~ior, of Cl1 1> This is called the Green's Function

Represr-ntation of the soluticn. We will use this form of the

GTI to address the questions of boundedness and asyrnptotics.

5. A Dircect, Approach to Boundedness

Th~r apT-oach used in the se.-ond section to show that pCA'<l,

suf f ici,-- for the- botundJdnc-!s of (1-7) is the one o- igi nail y

empioy-jd by thi author in ]53 arnd al s appears 2nr, 11 .. A

more d--rect method, however, was used by the aut-hor in [191.

and we now give this method

Supp.1se that pCA-D<i. Then there exists some natural matrix

norm Il . 11 such that 1lA1l < 1, Using IVI[ 0 to rppresspnt the

vector norm which induces thi - m;trix norm as well , we have

from (4.2) that

1:I

11\1 i 11 A-I V 11 + Aý *B v 1Kt =0

VrŽ_O. Thus

V < V 1; S IIA I:-AI;V PA I [ .n*3i Cl OK O Ot ct =

where • is a positive constant such that

11Vl / 1;V;; -< k

for al! nonzero vectors V. Such a k exists since all vector

norms are equavalent on Ri (12a. Summino the geometric series

and cole.-tino like terms aives

v i _- , + k- - A -

- aS n -- a. Hence IN li is asymptot.ically boturinFd,

and by the equivalence of vector norms so C7 ; iV 1K Ar

asymr. o-ic bound for IIV II is given by

0 rk'lhrl~i -'l - iA~i 7)

ir, t.erns of the a-norm. When the spectral norm of A is less

than unity the corresponding bound using that norm,

IIBIL/CI - UAII),

is gene-ally fairly tight.

Wv have shown that pEAD<1 suffices for boundedness. In an

analagois w;y we can show that if pk A))1I then there must exist

some V such that. liV I is unbounded. CA similar arproach has

w I-,

li. &.. .... . I

C 4. L-, ha h vr- t h;.t

,i Vo .i >_ 4 ,,' _ 0 1! Ao *F* , ý1 Vt

Cuslrng •he spe-:tral norm). Let > be ar. eigernvalue -of A such

that Ij', !=pCAD and choose V to be an elgenvec tor of A0

associated with the eigonvalue X. Then we have

llivn~l oI` 0 - 11 E1 A-'~*B'*V t -11V tt=0

anrd sin.-e the norm of the sum is at most the sum of the norms,

I'"• $ ' '• p "k r' ' 1• 1i A' -- r*1 1] 1A A' t i

IfI CI-i

c ~ ~ ~ /C ilA ý I lD

t =0_> r° Irl -~ i *II.C - IiAhn -'< - IAK:•

with F-=lV iV. If w• let.

C = IIE~i.CIIAUi - 1)

then c--, and

it V 1i f Ipo,.' •. ,X+ c tA1_ih j~x - clIAiIt' + ci

in term!-: of c. Now p'CAD=X -+E- for some e0, so

1IVr,, 1. ? 00 x , -c: (7x + C-)r + C

I c IOI + */ i '>0 .,X ÷c 'C ( ('5 1-)

If e iLu . ._t zero. we may now choose p to be gre;t.er thar,

c in or-ier to have liV W -- o.,1 Otherwise, the term involving e

will grow arbitrarily large, again giving lVV n 1

Agai n, we remark that the GTI (C1.1) has boundedness

properties similar to those of the linear system C2.2). In

the next- section we will derive improved asymptotics for the

indeter;ainate case where A has unit spectral radius. Before

dingr sc, let us collect the information above as a theorem.

THEOREM .3: Consider the iteration (C.1). If pCA7)<l then the

iteraticn is bounded for any V , and if p(AD>1 then there0

ii

exoists- a V i -hat the iteration is unbounded.0

:.. Improved A.yrmnptotic Eitimafes

From C5.1:) it, is immediately apparent that when pCAD>I the

divergence will , in general , be exponential . This again

leaves the case p<A)=I. As before, let us first look at the

case where A is of bounded type. Using the spectral matrix

norm, -4. F- gives

:< K + E 11-A.i; - 1lBI.t=0

where K Ž>C is a bound on PiA"v 1V Cwhich in genera] will1 0

depend on V ). Now since A is of bounded type, iAA Mll is also0

boundec, that is, ItAr'I1-K for some K>O. Thus

IIV II K + IIBtI 11 KI

t=0

_< + Kn1 2

showinc that the divergence is ric worse than li near.

If r(A-)=I but A is not of bounded type then it is possible

to show that tiAn hi=(Cr'Tn) Cin fact IIAr'll=©np-1 ) where p-<m is the

maximal degree of all nonlinear divisors of A associate-d with

urimodular eigenvalues), where A is mxm. Hence from C6.1:) we

have

IINV + 11 <:5 11 A ÷ ll •1 IIN II1 + 1: 11A l- t lB-181

t=0

and clearly IWV II diverges at most as OCn" 1 ). (Tn fact, it.

diverge- as (Cn P), and henr7e' in the worst case as Okhre)).

Hence, whern .F A-)=I, if the i tera,.ic-n divercrges then it. d, sc

A~ L-

THEORu 4: Consider C1. E-D and suppose that #CA = A and th;-

iteration is unbounded. If all Qigr-,vnaluoy C" A with unit

modulus are simple then the divergence is no worse than

1 inear. Otherwise the divergernce is no worse than polynomial

of order m, where m is the order of A.

7. Limit Iesi

Lot us ufainx FOA,R,V - to bn tho limit. se" ,.,r c not0

E131i of the iteration; that is, Ye7CA.FE,V D if there exist.s0

ar, in - i subsequeince ni .a such that V -- Y a-4 -ar- when

the in , taY conrdii tion in V . Twn t henrm-¶ fo 2 caw --. A'y frorr;(I

the mtý-ria] in [121.

THEoRI•m Q SUPPse that for some- V' te - "v is0

boundes Th-,en FC A. B, V D is rEo--;nnty compraz- sei arid

V -- ,FCA P V D -t• n--•.

PR)oF: GieTar-Jy the closure of - is simply F and so F is

closei. -re b~onzrdrýdess of fV > impl1ies t-ht F is also

7,-,unde. Her ce F i .z cnmpact Cby d-f a ni tioon,. by the

Bolzanc.-Weierstrass Theorem [2?], F is not empty.

Now V -- F means that infC OIV -Y' YeQF)-,O as n--M Q 12.

Suppose the contrary. Then we can find a subsequence of <V ",

indexed by, say, nCiD, which remains a finite distance from 7.

Now V " is a bounded infinite sequence and so by the

Rolzano-Weierstrass Theorem it contains a limit point. But

that pcrnt must be ir, F, which is a cor,tradiction. o

13"

THEOREr' ,-F SuppO:S-e that for somr V the sequence <V > s

bo u, "- •;,; ; 0 -C)CA.F-,V Ther7 FCA.,FV Ii p t tIvel v0 0"

invar aint

PRooF: Def ne

TCV) = A*V + B*V,-'iVI C7.1)

so that

V TCV)

is the iteration under consideration. Since Or-,, the

transfcrmation T is continuoLLS in some nei-chborhood of any

point. YrF. Fix Y and choo.se a subsequence nCib such that.

V -- Y. By the continuity of T, we have that.

TC V )--TCY)

"TCV b = V

.C) that

V -- JT y")

where mri =nCi) -4-. Hence TCYDeF, Ar, t r is ponai tIr vcI

irivariaa-,t. []

I r f • t., *. t ci ;-, f~rcrr the abv t~hat1 TY)F •'-e, ver Y•_-

and TCY-'i is defined. In other words, F\{O) is positively

invariant. Further topological results on discrete dynamical

syystemr- that can be extende-d to this cavp Cwith some care

taken near the origin) may be found in (14).

Suppose that Ye-C A, B, V ), so that V -- 4Y for some

increas: ng subsequence nCi). Since the transformation

T:UZ--ý[ of (7.1) has the property that

TC-V) = -TCV)

14

it f ci."o's that the se qu-nce gener-ee by -V is -V a ,

tendh -Y that is, -Y -7-A.F.- V r.. r. a s mi ar war W ',0

easy 4-7, SeE that -Yet-A,-5,V and Y-C -A.--V .0 (1

Ia, the case of a '--aI attractor, r 'A..V " A S 0

C. •i- the particu1ar A ano E unapJ on, S1der ai -or., l•,

case, at fo0lows from the above that the attractor Is

centrra ly symmetric. i.c. YEF implies -Yes, and that the- same

attract or is found when, A and B are replaced by -A and -B.

resEctZive4Fy. Some of the attractors d1 srl aveF- bv this

i terat on fc.r vmra ous c h ci ces fc A and B are shcJ. i r, the

A lcue- ; sea & a [4 l . i--res h fourd an anIal- I.-

exorese-Ior, for the attractor- focr a cer ta, set of case; c:

the f-rr ' Azo-. 0>? Csee [4]D. For the case A=O the forn of

Zh. a+raztor is also known IE, E 17>

, .- ,lti-d Theedorus iteration a.s in som,- wavb very

sim:" to the simpie inear s vtsm C7heor-e,:. 3z an. a e-i it

contains some very comTpiAcated structures CFigs. 2-57. It is

lik-ely that the attractors are, in, general, strange but nrot

always chaotic Csee [D; some computati-ional evidence to this

eAf ect has been provided by James HeyTran of t he Naval

Postgraduate School in his study of the use of chaotic

disczre - dynamical sys-tems, particularly the HNrnon attractor,

as psCdorardom number generators Cpersconalcz,,,munica':,

• " "i I I I I.

0 -r z: zr 7r~ :, Du oC t "G p 7eer, r&ý e C

d;f i ni t ionscof t h i t, crwr m chacs- &nd " f r a ct C se c ~' i

as dIf f: C cUl t to, pr ove amyt-hl rc s uh-s t-aart,~ Iye abouLt the c hao~ti.C

Fat1,ure C4C3Z th at r azt,ýr S bu, a s r, te & n I M JI manry

wel I. S-1-wn a rar ý ai 4.,- a z I, c.'..'S are st, ;. It a i~ "Ia y-t..C

ver1f ic ati On cof the o b ser-v e - qua'.i atatave fceatureýc cf the,

underi1 yl mg i teat ions arnd d.1 f fer er.ti al Systems. One aspect, Of

the GTT Which wc- feel is part-icularly relevant-1 to0 th1-e field is

t hc va -7t ar a o` aem-I- c al :v d. sti 1n,7 at t,,-a c- zr~ to*ýC be

f o und,-' n th - GT:- fo d. : ," - er en " cFoa ces of t-hac ma*,r aces A ard-

E;eve-. thuhtheSe tWo7 Ma-r aceSesetay rC-7epr e7-ten ac

pararm;et-s. the number- o f dal.Sta c t. attractors IaS St:EI

~~~~~~~~J, ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ S fr .L e . s... c nC .. .,...c .. , ý c-t e

mar Wi 'L IT-, 1 , 7-. 1 Further 4, Ftt Eo, a mn of- a has

~-rt 7e~r ranite C- ~ f j V b fi1 0;ur-eS C ted here

a n 71 r wal i rtvt research oni the qesao ca t heC

rnumber of di -,t irict att~ract-or s wh.-ch can belorn- to single

M rtan 1 - the wc, ,C bY GleiclkC7 c -T:. Eean m -i- an-,

Wac LYs , Z- have;- hrrve Pc T-117,1 I vat-e In rif or 7 th! S

author .

Acknowledgements: This research was supported by the Naval

Postgrcaduate Rcoo esearch Council Foundat"ion.

LI)

// //

I

q-4

// -

/1 �/

�N.

- - q�NII I I I

LI)

CN N

U! I)

iit

Ln

~~ A

Ln Ln Ln ,mC

I I -� I

r

'eq Li-r4* �./

/ :ii,(I

<itt

fij aN L

7./. 1 L-'1�- xr.* - *1 p#4* x ��&/%? rI- �;'z-'.

V C.�!f* C- * . -�

�Ad it

#( �t*.t�% * V

/

:'� 4J \ K-. -� NI 9.6C

.rC **'� r- . I La

a

liD

q I I I

In '.4 In In 54 A- '-4

I I

A' 1~ -40I

lie

17~

;-Vcc

/ C'

/ r

II

I II I _ _-_

I I ~ NI

I II

Lr

~ a

Cz'

14

m CD ) C

As ymn I oti c F;= I,: c~ur of &71 l- at crI OfS f Li n;-rr arid N,-n-.1

E)i f f e,-ent ial and Di f f e rnc-.cc :- iju, tjion7= Cf f Ice of N~ztv-ý1

Rosearh GZ

I I ByF' (-.a Cr I. R~. A Ht F o r o f Mo. * e~ma tc s Pr irn e 1on 195-:

r-; F-viaI PL iip J. m 'Th c-,- -

Me ta-C &ac o~ s Hedrick Lecuur-Eýý Notes

L 5- Z -7 Lai s . Fr i1ip' . Tho,,rrtrzý GrayF-~sce Car, Har-court-

FEaFCE C-Vr7Vii

P-; Davis. F1-'iiiip J. :pi.-t 10 5 : FT 7 ro Thkeod071-c'-7,5 c C c Cr, .7

andQ Bý:7tlieLt icxpect~ed January igcý

t& 1-i Aa 7-, ,7- Tam Th o' lU 7-t t C hew C-, zl Qr.M- Vi C CZ

StrarLe- At tractorsc That are n~ot Chaoc;tc, Phy-sica IKý: E1-l

110i Hlawka. E. G-Lez~chverte-Ltiune unT, dT-UT&S-rZl&

Monats'-,efte fujr Mathematik. 89, 19SO

[11) Krcraý , Husscyin bt ff er..'n aZ and Differencze Eqiat orts

throvej:, Compi.Uer F-p'f-'ments; (Ernd ed.-), Springer-Verlar7 i98C

-13 La'ý,-ca~ter , Pr-i -r- ani M'i ron Ti -m;-netsk y The Theory)F Of

Mlt TiC-'-ý (Enri Pri.)Acdra rs 19-R

L-J . P, The 5tub-L Zity and Controfl of Dt.SCrre&

F~r~z~e. 5.~irgerV~jag I~~

f14 Latian, V. R. To~k-zrdcz a Ur'tiic-i Thor-r c,, of

Fuic- t t on for DLvScreze I terait ors.s, Ph. D. T1hesi. S Prown

llý; L'3-RrJffc-ry J. The eez-~ Th-eodor-U. Jteratzon,

Ph. 1. Thes is , 6;- cgwr Un iver sitAy 1 99,

Lrl6l Le7ade-r- Jeffery J. Lmint Oz-btt o! a Power lteprat tort for

Dom~-,c~r Ea r.=v~iabe Frob~emq., Applie-d Mathematics Letters,

VcJ.4 No- 4

[175 L-:?aci.r. Jeffery J. Foumer lterattortc u'~~ the Dontiricult

r -r,oLv rhe~ NS Teo-hnic~lU Report, submitted

~1;L~, .74ffery -T. A Weaý,y I'*T '- It ercit ton in n

A-'i~~1 Mth~~t'csLetters. V0l.4 No.4

d i" fz- f Y J3. bo-undedne--,s and Asymptot-tcs of a

K'.~riyUexa-r~ ~vM~urtaa~i3 of Mathematics, tLo appear

2- L I fjf ýr Y .Chao's k n Eu Z'r' 5 Hk--t hod for t hes

o. O, ~ e. c( Crd-znarv 1, ?*f!feren t i zI Eqluations,

NF nh 2h c; I tir~r .sr-,mi t t-e.3

~2A1M~1 ir f S. r-ta 7tf er- Equa t L ons, W.A. Pe~n iamirn

[123 Pracf-. G. Faley Mu~tivaxriabl~s AntaLy5sts, Springer-Ve;rlag

1 -cE4&

(E~41 Tc- -fel . E. Eitne Enc f.d.-r Qcdwrcrele

M 't1 . hys-. Smt.rrihe6. lq55ý

r -ýJi ns, ~Phr G C ICbo F'~ fvr c at tin c a- S cr,c Cc'

[ ~ ~c~ ~ or ~C, ApjA,, N,7 rci Io.~tr~

C'y.ira'~a? v~tmc r~c Clto~~,Spr inq- -Vk~r-1~ I a

IL

I = RUTi R U

D~r etcF t 2

fen s-; T ezC I , f Z-r-In at i Or, C ent1e r

Canneror. S"tation

Alexancr-aa, VA 2-=''714

Li brarv 2

Nava'L Fzxstoradjat~e S~chooil

Mzimterc.-v CA 93Z3~4-?

Pesearc-. Off'ice2

Code- 6ý

NAva-, F.trdae Sr-hool-

Momt~er>-.:.- CA 34

If)-. Jefr~ery j. Leader, Crode MA,-LE

Navzs. ---r~uteShoc,

1~o~t~re',CA C--7,-43

DR. ~c`arci Frankde, Code MAY.-FE

Naval Pcstaraduate School

Mcrmtere- -, CA QZ-943

LT James! H-Eyrrarn, CODE Z?9

NaVal P~itIC:7adUatC- -SchOol

LT Ton' Forntina, ZODF MA1

Naval Postgraduate School

M-:n terey, CA C4

Dr. Todd A. Rovelli1

Divisicn- of Applied-1 1Mathemat.-cs

Br own L'ni veri~ty

Providence, R! O2'ijla

Divisicn of Applied Mathemnatics

ronUrnivc-rity

Provide-nce, RI O2,:,Er

Dr. P'aul Frankel -

De-Ptart--nr~ati of Mathema.tics

Un ive rs. `-y o f Sout h err, Ca 1i f or ra

Los Ana,-- es, CDA 1-,3&-

Dr. Ismojr Fi-scher, CODE MA/ýFl1

Naval P.Dstgraduate School

Mo:ntere.., CA 9S4ý