map models for the emergence of ordered states out of chaos

Volume 133, number 7,8 PHYSICS LETTERSA 28 November1988

MAP MODELS FOR THE EMERGENCE OF ORDERED STATES OUT OF CHAOS

PaulE. PHILLIPSONDepartmentofPhysics,Box390, UniversityofColorado,Boulder, CO80309-0390,USA

Received18 May 1988;revisedmanuscriptreceived19 August 1988; acceptedfor publication31 August 1988Communicatedby D.D.Hoim

Investigationis madeof onedimensionalmapsx,,~,= (I —r)M~(x,,)+rMq(x~).M~andMqareassumedto bechaoticattrac-torscharacterizedby p andq critical pointsrespectivelyandr is a controlparameterboundedby 0 ~ r~1. Fora particularclassof maps,M~(x~)=cos[ (a+ I )O~]{x~=cos(O~)},multiplicitiesof periodicorbitscancoexistover a rangeof thecontrolparam-eterwhich model a dynamicsfor theemergenceof orderedstatesout of chaos.

Studiesof chaosdisplayedby dynamicalsystems backgroundwherethe orbits areunstableto a con-and by mapswhich model thesesystemshavebeen trol parameterregionwithin which theorbitsaresta-concernedfor the mostpart with how chaosarises ble attractors.When orbits becomeattractorstheythrough successiveseriesof instabilities [1,2]. The appeareithercompletelyor almostcompletelyshedconverseproblemwill be addressedhere:how can of backgroundstates.Backgroundstateswhich maythe interactionof chaoticattractorsleadto predic- persistarelocalizedchaoticattractorswhicharesmallably orderedstates?The aim of the presentwork is in extensionandconfinedto parameterregionssmallto demonstratethat the dynamicsof the emergence comparedto the regionof stability of the periodicof orderedstatesoutof achaoticbackgroundof states orbits. Whenpresenttheyfunction dynamicallyasacan be modelledby suitablycontrivedone dimen- limited amountof noise in a basicallyordereden-sionalmapscharacterizedby multiple critical points. vironment.Therequirementfor this stabilizationofThe iteratesof a onedimensionalmap eithercon- periodicorbits will be shownby exampleto bemm-vergetoperiodicorbitsor toa chaoticattractorchar- imally the interactionbetweentwo or morechaoticacterizedby a wayward distribution of iterated attractors.It is the interactionbetweenchaoticat-points.Whichoccursdependsupon the value of a

tractors,asmeasuredby thecontrolparameter,whichcontrolparameter.On theotherhandperiodicorbitsleadsin thefollowingschemeto stabilizationof states

of all periodsexistin thechaoticregimebuttheyareunstable[31.Thisimplies that if thecontrolparam- representedby attractingperiodicorbits.

Themodel dynamicsof the transitionfrom chaoseteris suchthat iteratesconvergetooneormorepe-riodicorbits it is possiblethatsomeorbitswhich are to order,andvice versa,is baseduponconstructionunstableoutsidealimitedparameterrangearestable of model mapswhoseiteratesarechaoticin the ab-within the range.Suchperiodicorbits will be iden- senceof coupling betweentwo or moresuchmaps,tified with coherentstatesandthewaywarditerates butfor which thecoupledsystemhasstableperiodicof a mapin its aperiodicregimeasbackgroundstates. orbits. Considera one dimensionalmappingM( r,Theemergenceof orderout of chaosis envisagedas x) characterizedby a singlecontrolparameterr suchstabilizationof theseperiodicorbits as the control that a variablex assumesdiscretevaluesaccordingparameterprogressesfrom chaoticthroughnon-cha- tox~+ = M( r, x~),n = 0, 1 .... Themappingwill beotic regimes.The transitionfrom chaosto orderis expressed.as the sum of two mapsM~(x~)andinterpretedasthetransitionofpre-existingspecies— M~(x~)coupledby the controlparameteraccordingidentified with periodicorbits — from an aperiodic to M(r, x) ~M(p, q) where

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Volume 133, number7.8 PHYSICSLETTERSA 28 November1988

~ + 1, andthe generalmapping(1) is assignedthe=M(p, q)= (1 —r)M~(x~)+rM~(x~) specific representation

(0’~<r~<l) . (I) x~~1_—zcos(0~÷1)=M(p,q)

ThemappingM~andMqare sochosenthat: (i) they = (1— r)M(p) + rM(q),possessa=p, q criticalpointsc~respectivelysuchthat M(p) =cos[ (p+ 1 )0~], M(q) = cos[ (q+ 1 )0~IM~(C1)=0,ji, 2... a, (ii) MpandMqare chaoticattractors.The first condition implies M(p, q) (p<q) . (3)evolvesfrom a mappingof ~ocritical pointsat r= 0 M( 1) is the exhaustivelystudied quadraticmapto a mappingofq critical pointsat r= I. Thesecond [1,2], Ax~/2— (,~.—2)/2,at theparameterlimit 2=4.conditionconstrainsiteratesat the control param- theiteratesat 2 = 4 arechaoticandgivenanalyticallyeterlimits tobechaotic.Yet stableperiodicorbitsof by x,, = cos(2”0~). M( 3) is the cubic map originallya particularlyrobustnaturecanemergebetweenthese studiedby May [5], 24 + (1 —2)x~,alsoat thepa-parameterlimits. In generalif a onedimensionalmap rameterlimit 2 = 4. Theiteratesat A = 4 are chaotichas a critical points the maximum numberof pe- andgivenanalyticallybyx~= cos(3”0~).If thesetworiodic attractorsgeneratedby the map is equalto a chaoticattractorsarecoupledaccordingto (3) the[41. Theexistenceof multiple attractorsimpliesthe asymptoticiteratesx,, definedby (2), of M( 1, 2)existenceof multiple basinsof attraction.Foralmost asa functionof r areshown in fig. 1. At the param-all initial conditionsx0 thereareregionsof the con- eteredges,r= 0 and1, M( 1, 2) reducesto M( 1) andtrol parameterfor which iteration according to M(2) respectivelyso that by constructionthe iter-x1 =M(r, x0), x2=M(r, x,) ... canconvergeto one atesare chaotic.As a result of the couplingof theseof m ~a periodicorbits a ~ of period lengthL1, orL-cycles, whoseelementsare periodic points an = 1, 2 ...L~.The pointsof a~ subsequentlyiterateamongstthemselveswithperiodL~.Thesepointshavethe properties

x~(x0)= lim x~(x0)ea~-’~,j=0, 1 ... (m~<a),XOD

a~/~1=M(r,a~/~),a~L~—a~’>. (2)

To which of the m possibleorbits the iteratescon-vergedependsuponx0. Thesetof all pointsx0 in thespaceof initial conditionswhich convergeto a ~ de-finesthebasinof attractionofa~ As aconsequenceorbits,generatedaccordingto (1) cannotonly arise 0 I

in a multiplicity dependentuponp andq, they willadditionally competeas attractors for almost all Fig. I. Iteratesx0,,(x0) versusr of M( 1, 2) definedby (3) ac-

points in the spaceof initial conditions.The set of cordingto (2). r scaledivided into 640 increments.For eachinitial conditions(that is, the closureof this set) valuetwentyvaluesof x0 werechosenin stepsof 0.05 over the

spaceofinitial conditions.Eachx0valuewasiterated1000 timeswhich convergesto eachparticular attractor is its (x,000(x0)...x~,,(x0))subsequentto whichwereplotted thenextbasinof attraction. 200iterates.Theordinatefor eachvalueof r representsaplotof

The simplest realizationof the requirements(i) 4000 points.Thus, any interval in theaperiodicregionappears

and (ii) aboveis basedupon homotopiesbetween almostsolid, while in theregionof stability thesepointsare

polynomialsofdifferentdegrees.M~(x~)andM~(x~) dundantly periodic points. The two horizontal lines betweenr,,10=0.2 and rm=0.6 are the robust 2-cycleperiodic pointswill bechosenaspolynomialsof ordera+ 1 in x,, such a, = 0.309017(topline) anda2= —0.809017(bottomline).The

that Ma(x~)= cos[(a + 1)0,,] where x,, = cos(0,,). arrow indicatesthe counterexamplediscussedin thetext of aThespaceof initial conditionsis — 1 ~ x0 = cos(Os) period2 orbit whosepointsvary with thecontrolparameter.

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Volume 133, number7,8 PHYSICSLETTERSA 28 November1988

maps by the control parameterr oneobservessta- abilization of a period 2 orbit whosepoints, (a1,a2)=(—0.809017, 0.309017), commence atrmjn=0.2 andterminateat rmax=0.6. Readingfromright to left this 2-cycle mergesinto the predomi- 3

nantlyM( 1) attractorat rmjn by the familiar root ofperiod doubling, and readingfrom left to right it X0~mergesinto thepredominantlyM(2) attractorat rmax I

also by period doubling. The period doubling sce-nario is ubiquitous,of course,to one dimensionalmaps.Thedifferencehereis thatwithin its regionofstability 0.2~r~0.6 this 2-cycle is independentof 3

the control parameter.This is reflectedby the fact - I0 I

thatthe two linesat a1 and a2arehorizontal.Becausethe pointsof the period2 orbit retaintheir quanti -_______________________________________tative integrityoverits rangeofstability it is thepro- 3 btotype of the class of L-cycles which will be — - - - ______

characterizedasrobust.RobustL-cycles aredefinedasstableorbitsof periodLwhoseperiodicpointsareindependentof the control parameterovertheir re- —- _~_ - - ______

gion of stability. The analysisto follow will dem-onstratethat the causeof this featureis that ~anda2 are unstableperiodic points of both M( 1) andM(2) separately.This2-cyclecanbeviewedasa pre- —~ I — — - —

existent object,unobservablein the aperiodic re-gionsfor r< rmn and r> rmax, but stabilizedby cou-pling oftheattractorsby thecontrolparameterwhen — —- ~ - - _____

— Irmn<~r~<rma,,.All points in the spaceof initial con- -I

ditions are attractedto this uniqueperiod 2 orbit Xo

within this region of stability. The feature of ro- Fig. 2. (a) Iteratesx0,,, (x0) versusr of M( 1, 6) definedby (3)

bustnessis distinct from the usualcasewherepen- accordingto (2).Detailsof theplot arethesameasin fig. 1. The

odic pointsvarywith the controlparameterwithin horizontal line marked“I” is a robust 1-cyclewhosepoint istheir regionof stability [1,21.An exampleis shown a,= —0.5. The horizontallines marked“3” indicatethecoexist-

in fig. 1 wherea 2-cyclearisesby tangentbifurcation ing robust3-cyclewhoseperiodicpointsfrom top to bottomarea,=0.7660,a2=0.l736,a3=—0.9397.Thearrowindicatesthe

at r= 0.8 (indicatedby an arrow) followedby a pe- periodI attractingorbit discussedin thetextwhichcompetesasnod doublingscenarioon a reducedscaleinvolving a sourceofnoisefor initial conditionswith thetwo robustattrac-

thetwo allowablebasinsof attraction,similartothat tors. This attractormimicstheperioddoublingscenariocharac-

ofthecubicmap [61. Fig.2adisplaysx,,, versusr for teristicofthequadraticmap[1,2]. (b)x(x0)versusx0for M( 1,the mapping M( 1, 6) to illustrate competitionbe- 6) definedby (3) at thesuperstableparametervaluer,= 0.2222

for thecoexistentperiod1 andperiod3 robustcycles.Thesecyclestweenmultiplerobustattractors.Thefour horizontal areindicatedasin (a).Thebasinsof attractionaresuchthat alllines indicate a 3-cycle coexisting with a 1-cycle, initial pointsx0on theabcissaunderthe“1” lines areattracted

Readingfrom left to right, bothattractorsbegin si- to theperiod 1 orbit andall initial pointson theabcissaunder

multaneouslyat rm,,, = 1/9 andbecomeunstablesi- the“3” linesareattractedto theperiod3 orbit.multaneouslyatrmax= 1/3.Fig. 2bis aplot ofx,,,(x0)versusx0 which showsthe competitionof thesetwo of initial conditionsandthe ordinateis x, whoseattractorsfor the spaceof initial conditionsat the valuesare confinedto eitherthreelines (period3)superstablepoint [1] r~= 2/9 at which the slopeof or a singleline (period I). Almost all x0 upon it-theM( 1,6) mappingis zero.Theabscissais thespace erationwill be attractedto eitherof thesetwo at-

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tractors. Thus, for examplethe point x0= —0.4 is 4 aattractedto period 1 andthe point x0=0.2 is at-tractedto period 3. The exceptionalpointsare theunstablebasinboundarypointswhich separatethecontiguousperiods 1 and 3 basins of attraction. 2

Computationalevidenceis that thesebasins,with theexceptionof the basinboundarypoints, cover the x

4entireinterval. The computationsalso indicatethatthe topologyof thebasinswhile nontrivial is never-thelesssimple, that is, not fractal [7,8]. The com-petitivenatureoftheseattractorsis measuredby the

bproportionofpointsin thespaceof initial conditions 2

which are attractedto period I and the remaining -

portionattractedto the period3 orbit. In thiscase 033 100

a valueof x0 chosenat randomhasapproximatelya62% chanceof being attractedto the period I orbitanda 38% chanceof being attractedto the period3 . ..._~ ~4 . —. borbit. Theseproportionschangeto a smallextentwith

4r within the region of stability. More importantly, --~ -—

while thesetwo robustorbitsmaintaintheir integrity2

over the parameterranger as shown in fig. 2a, the —, I I I.

numberof availableattractorsincreaseswith r. As aresulttherecanexistadditionallycompetitiondueto x0~ 4 - —

residual“noise” in the formof non-robustattractorswithin the samewindow of stability.Fig. 2aalso in-dicatesthat for M( 1, 6) sucha period 1 orbit arises . ~.. — . -

throughtangentbifurcationat aroundr=0.298(in- - b

dicatedby an arrow) which subsequentlycompetes - — — - 2 -_4

with the two robust attractors.A more transparent - — -—I I

exampleis furnishedbyM( 1, 27) which featuresthecoexistenceof threerobustcyclesof periods1, 2 and4 with residualnoise.Fig. 3a is a plot of x~,,versus Fig. 3. (a) Iteratesx~,(x0)versusrofM(l, 27) definedby (3)r over the regionof stability showing the sourceof accordingto (2). Detailsof theplot thesameasfig. I. Parameter

rangerestrictedto rm~(1/30) ~ 1/10),theregionofsta-the noise (denotedby a andb) aroundthe super- bility ofthreecoexistingrobustcycles.Thehorizontalline marked

stablepoint r~=0.O666.Fig. 3b is a plot of x(x0) “l”is theI-cycle, thehorizontallines marked“2” indicate theversusx0 at r~whereinterruptionof the basinstruc- periodicpointsof a 2-cycleandthehorizontallinesmarked“4”

tureof the threerobustattractorsby the two attrac- indicatetheperiodicpointsofa 4-cycle.Three(non-robust)at-

tons (a) and (b) appearas squaresor rectangles. tractorsin this parameterrange,which functionasnoise,arein-dicatedby a,bandc. (b) x,,~(x0) versusx0for M( 1, 27) defined

They exemplify coexistingrelatively small chaotic by (3) atthesuperstableparametervaluer,~=0.0667showingthe

attractorsreferredto abovewhich disruptthe pre- influenceof noiseupon the basinsof attraction for the robustdominantlyorderedenvironmentofthethree(in this periods1, 2 and4-cycleattractorsindicatedasin (a).Twocorn-

case)robustattractors. petingfractalbasinsofattractionappearasrectangles,indicated

Computationsof x~,,versusr andx versusx0 for by a andb correspondingto thesamedesignationasin (a).various (p, q) combinations within the range[1 ~p ~ 5, 2 ~ q~210] reveal that while different exist periodicpointswhich are independentof thechaoticattractorpairsexhibit differingpersonalities controlparameterwithin their rangeofstability, (2)theyall displaythreecommon features:(1) robust- coexistence— the multiplicity of periodicattractorsness— foraninfinite setof (p,q) combinationsthere areall stablewithin thesamecontrolparameterrange

386


~ (p,q) pairofcriticalpoints, q=[(p+l)’-+l]rn/k~(p+2)(3) competition— coexistenceimplies necessarily /2~k(p+1 )~

that all robust attractorscompetefor points in the ~ =CO5~ + 1 “-.— 1spaceof initial conditionswhosebasinsof attraction /

becomerapidlyofcomplexandrich topologyforeven q [(p+ 1)L_ ] rn/k—.(p+2)modestvaluesofp andq. Thereis additionallycorn-petitionduetoresidualnoiseintheforrnofnon-ro- (n=0, 1,...(L—l), in,k=1,2,...) . (5)bustattractorswhichcanexistwithin thesameregion Therangeof controlparametersforwhichtheperiodof stability. Thisis permittedbecausethenumberof L orbits are stable,the superstablevaluer5 andthecoexistentL-cycle attractorsis in generalless than stability width A aregiven bythe numberof available critical points.Thesefea-turescanbe placedon a formal footingby consid- r — r — r — ______

ering a set of L points (a0, a~...aL_).Thesepoints m,n p+q+2’ p+q+2’ p+q+2’will be said to definea robuststableperiodL orbit, 2or robustL-cycle, providedthey comply with three 4= r,,~— ~ = p+q+2~ (6)correspondingconditions:robust condition: Proofof results (5) and (6) is given in the appen-

dix. Eq. (5) showsthereare two orbits, (+) anda ~,=M (a )=Mq(a,,), n=0, 1 ...L—l ; (4a)

(—), for a givencycle lengthL. ForeachL-cycle theperiodic condition: equationrelatingq to p encapsulatesa selectionrulea = a (4b) which determineswhich (q,p) pairof attractorscan

+ L ‘ give riseto this robustL-cycle. Theserulesarerather

stability condition: complexsincetheyimplicatetwo integerswhich are

L — I (~j~- ‘~ so constrainedthatonly (m, k) pairsarepermittedIA(r)l = 2(a,,) ~ 1, 2(a,,)=~~j~

1) for which q is an integerfor a givenp. Secondly,k— (4c) determinesthevaluesof thecyclepointsa,, foragiven

numberof critical pointsp anda givencycle lengthWhen(4a) is insertedinto (1) thenx,,÷~=a,,~in- L. As k runs overall integersit will generatea mul-dependentof r over the entire interval. Since (4a) tiplicity of distinct L-orbits for the (+) and (—)

and (4b)combineto makeeacha,,a periodicpoint casesindependently.In generalthemultiplicity of L-of M of (1) it mustbe an attractingfixed point of cyclestends to grow algebraicallywith p and geo-M1L) theLth iterateof M. Theconditionfor its sta- metricallywith L. Applicationof the selectionrulesbility is that the slopeof M(L) evaluatedat a,, must showsthat thereis a greatdealof degeneracyin thatbelessthanunity. Eq. (4c), which is a statementof many (p, q) pairsproducemapsM(p, q) charac-this requirement,follows fromthe chainrule fordif- terizedby the samerobust cycle structure.Eq. (6)ferentiationof maps [1,21. Thetwo valuesof r for showsthat the rangeof stability is dependentuponwhich the equalityholds, A(rmj,,)I = I~4(”max) I = 1, only the numberof critical points andis indepen-definetheparameterrangeof stabilityof theL-orbit dentof orbit length.As a consequenceall robustor-of extensionA= Tmax — rmjn. Within this extension bits which are allowedfor a given (p, q) pairhavethereis alwaysa uniquesuperstableparametervalue the samerangeof stability. Additionally, while no-r5 such that 14(r5) =0. Application of conditions bustcycleperiodicity andmultiplicities tend to in-(4) to theparticularmapmodel of (3) resultsin the creasewith increasingnumberof critical points, infollowing analyticalexpressionsfor the robustL-or- compensationfor this complexityis the featurethatbits andfor thecombinationof critical points (p, q) the parameterrange of stability decreasesas 2/which generatethem, (p+ q +2). Someof thesecharacteristicsare illus-

/2mk( + 1 )“\ tratedin table 1 which lists the coexistentrobustL-I =cost\( + l)L + i)’ cycle structureof maps M(p, q) for p= I and

p 2~q~50.

387


Table 1RobustL-cycle structurefor mappingsM( 1, q) of eq. (3). Entriesareof the form n (L), whereeachn(L) denotesfor a givenq thenumbern of distinct cyclesof length L. For examplethemappingM( 1, 48)) givesrise to one 1-cycle,two 4-cyclesandtwo 8-cycleswhosecommonregionof stabilityis rmi. = 1 /51 ~ r~rma. = 3 /51 from (6).

q n(L) q n(L) q n(L)

2 1(2) 19 1(5) 36 1(1), 1(6), 1(12)3 1(1) 20 1(11) 37 1(2)4 1(3) 21 1(1) 38 2(10)5 — 22 l(2),l(lO) 39 1(l),l(3),l(6)6 1(l),l(3) 23 1(6) 40 3(7)7 1(2) 24 1(1), 1(3), 1(9) 41 1(5)8 1(5) 25 1(3) 42 1(1),l(2),l(3),1(4),1(12)9 1(1) 26 1(14) 43 1(11)

10 1(6) 27 l(l),l(2),1(4) 44 1(23)Il 1(3) 28 3(5) 45 1(1)12 1(l), 1(2), 1(4) 29 1(1) 46 1(3), 1(21)13 — 30 1(1),3(5) 47 1(2),1(l0)14 2(4) 31 2(4) 48 1(1),2(4),2(8)15 l(l),1(3) 32 1(2),1(3),1(12) 49 1(6)16 l(2),l(9) 33 l(1),1(3) 50 1(26)17 1(2) 34 1(18)18 l(l),l(3),l(6) 35 1(9)

The identificationof periodicorbits generatedby parallel characteristicsof order—disorder phasemapswith coherentstatesandaperiodicorbitswith transitions.

backgroundstatesmay provide interpretativedy- Theestablishmentoforderimpliesthe impositionnarnicalmodelsfor physicalsystems.If oneconsid- of ruleson anotherwiseundefinedsituation.Suchisens robustorbitsfor relativelylargep andq, whatis the basis of gamesand their applicationto evolu-observedis that the transitionfrom chaosto stable tionary processesin nature[9,10]. The rulesof therobust cyclessymptomaticof order becomesvery presentmapgamefor the emergenceof orderoutofabrupt, andthe clearly delineatedperiod doubling chaosare the conditions(4), andmostimportantlytransitionpictureapparentfor p andq small is re- the robust condition (4a). Polynomialmaps wereplacedby sharpboundarybehavioursuggestiveof a chosenbecausetheir simplicity allowsanalyticdem-phasetransition-likeemergenceof orderoutof chaos. onstrationaccordingto (5) and(6) in conjunctionWhile parallelsbetweenphasetransitionsandperiod with theappendix.Moreelaboratecompetingrobustdoublingare understood[11. Fig. 4ashows~ ver- periodiccyclestructurecanbeachievedby couplingsus r for the mapping M(8, 201) which demon- None dimensionalmaps Mk, k=l, 2 ...Nwherenostratesa very sharptransitionfrom chaosto a single two mapshave the samenumberof critical points.period 105 attractordevoidof any apparentperiod L-cycleswouldberobust forarbitrarycouplingspro-doublingscenario.Fig. 4bis anenlargementshowing vided that the cycle points comply withthe transitionfrom chaosto order at rmj,,. Thetran- a,,+ I = Mk(a,,) for all k. In order to model the dy-sition appearsasa condensationof pointswhich are namicsof the emergenceof orderedstatesout ofessentiallychaoticallydistributedwhen r< rm,,,to the chaosthecouplingsmustbe soconstrainedsuchthatorderedsequenceof 105 points for r> rmi,,. Figs. 3a theN mapshavecommonchaoticregionsandall L-and 3b show how competingchaoticstatescanco- cycle orbits havecommon regionsof stability. Suit-exist within a predominantlycoherentstructureof ablecoupling of threeor moremapswould featurestatesidentified with robustorbits: Figs. 4a and4b a multiplicity of two or more regionsof stableat-illustrate map modeling of almost discontinuous tractors arising out of a multiplicity of aperiodictransition from chaos to order. Thesebehaviours backgrounds.Similar considerationswould applyto

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I a Appendix

Eqs. (5) and(6) arethe periodicpoints,themappairingsin termsof thecritical pointsand rangesofstabilityfor themapmodel of (3). Theseresultsfol-low from applicationof the conditionsof (4). The

X solutionsof (3) at the parameterendpointsare

x~=cos[(p+1)~00], r=0,

=cos[(q+l)~00], r=l , (7)

and the slope of the mapping is 2=dx~~1/dx,,- II I = — [sin(0,,) ] — ‘dx,,~ /dO, so that conditions (4)

.03 rmin rmox .06 become:robustness:

I b a,,=cos[(p+l)~00J

=cos[(q+l)~00] foralln; (8a)

I n=0

-I, I I +r(q+l) sin[(q+1)0,,]}/sin(0n)~l. (8c)0378 0379 .0380

Eqs. (8a) and (8b) are satisfiedrespectivelyby:robustness:

Fig.4. (a) Iteratesx~(x,,)versusr ofM(8,201)definedby (3)accordingto (2). Details of the plot are thesameas in fig. l, (q+ I )0~= — (p+ I )0~+ 2xm (9a)exceptthat eachx0 valuewasiterated1 500times.The horizontallines betweenr,,,~~=

8/2l1 ~0.0379 and r~5=10/21I ~0.0474 periodicity:

from (6) is a single period 105 robustattractorwhich is pro-jectedsharplyout of theaperiodicregimeat r,,,~,andsubsumed 2nkin aperiodicnoisejust as sharplybeyondr,,,,~,.The absenceof 0~= + 1 \L+ 1 (9b)gradualtransitionfrom orderto chaosparallelsthedynamicsof -‘ —

a phasetransition. (b) Close up of a boundarybetweenorder where mandk are independentintegers.Substitu-andchaos:iteratesx0,,(x0) versusr ofM( 8, 201)asin (4) except . .

restrictedto a rangeoft closeto r,,,,,= 8/211~0.0379 separating tion of (9a) into the first equalityof (8a) resultsintheaperiodicregionfromtheperiod 105 attractor.Detailsof the theexpressionsfor theperiodL orbit points (5). In-plot arethesameasin (a). sertion of (9b) into (9a) results in the associated

selectionrulescouplingallowedpairsof critical pointcompetingrobustperiodiccyclestructuresgenerated mapsp and q for a given periodL orbit. Theregionby suitable coupling of maps of higher of stability of coexistingL-orbits is foundby substi-dirnensionality. tution of (9a) into (8c) sothat, for stability,

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Volume 133, number 7,8 PHYSICSLETTERSA 28 November1988

A(r)J=J[(l_-r)(p+l)-—r(q+1)]’I References

r “LI sin(O,,+1)\xl (11 in (0,,) )= 1] ~ I , (10) [I] H.G.Schuster,Deterministicchaos,anintroduction(Physik

L “=0 5Verlag, Weinheim, 1984).

12] R.L. Devaney,An introductionto chaoticdynamicalsystemswhere,with theuseoftheperiodicitycondition (8b) (Benjamin/Cummings,MenloPark,1986).

the termsin the productcancelin pairsto result in [3] P. ColletandJ.-P.Eckmann,Iteratedmapsin theintervalunity. The rangeof stability,fixed by Al = 1, is sat- asdynamicalsystems(Birkhäuser,Basel,1980).

isfied by the two conditions (1 — r) (p+ 1)— r(q+ [4] J. Guckenheimer,0. OsterandA. Ipaktchi, J. Mat. Biol. 4l)=l at r~_rmjn and (l—r)(p+1)—r(q+1)=—1 (1977) 10!.

at r=rmax. Thesuperstablepoint, fixed by Al =0, is [5] R.M. May,Ann. N.Y. Acad.Sci. 316 (1979) 517.

satisfiedby (1 —r)(p+l)—r(q+ 1)=0at r=r5. The [6]J. TestaandG.A. Held, Phys.Rev.A 28 (1983)3085.resultsare (6). The independenceof cycle lengthL [7] 0. Grebogi,S.W. McDonald,E. Ott andJ.A. Yorke,Phys.

ontherangeof stability fora givenp andq is because Lett. A 99 (1983)415.

all termsin (10)aremultiplied by a commonfactor. [8] P.E.Phillipson,Phys.Lett. A 128 (1988)413.[9] M. EigenandR. Winkler, Laws of thegame(Knopf, New

As a resultall robustorbits allowedby theselection York, 198!).rule (9a) coexistwithin the sameregionof stability. [10] D. Farmer,A. Lapedes,N. PackardandB. Wendroff, eds.,

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istenceis implied by the propertyof robustnessforthe model (3).

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map models for the emergence of ordered states out of chaos

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