Volume92A, number7 PHYSICSLETTERS 29 November1982

FLUCTUATIONS AND STRUCTUREOF ATFRACTORS -

SIMPLE TESTS ON THE HENON MAPPING ~

K. STEFANSKIInstituteof Physics,NicholasCopernicusUniversity,ul. Grudziqdzka5, 87-100Toruñ,Poland

Received19 August 1982Revisedmanuscriptreceived21 September1982

Resultsof testson aninfluenceof fluctuationson a structureof a singlestrangeattractoraswell ason two coexistingattractorsperformedfor theHénonmappingarepresented.

Fluctuationscausedby smallrandomperturba- Figs. 1—3 presentingthe succeedingenlargementsattions arepresentin any realsystemandtherefore show whathappenswith theHénonattractor(partsa)they are a subjectof interestof linearandnonlinear whenperturbationswith different valuesof the dis-dynamics[1]. Relativelymanypaperswere published persiona are added(partsb andc). Like in ref. [10]e.g. on different aspectsof their presencein quantum- onecanseethat the invariancewithrespecttochangesopticalsystemswith simpleattractors[2—6].Recent- of thescalewhich is observedfor the “pure” attrac-ly someinvestigationsof suchakind on dynamical torbreaksdownin thepresenceof perturbationsatsystemsexhibitingchaoticbehaviourwere alsoper- thedistancessmallerthan approximatelya.formed [7—9].Theymostlyconcernedbifurcations TheCurry attractorsarepresentedin figs. 4 and5.in mapsof aninterval,butpaperson morethan one- Like in theprevioustest,partsa showthe attractorsdimensionalsystemsalsoappeared[10]. withoutperturbationswhile partsb and c showthe

We presentresultsof two simple testsof sucha sameattractorsin thepresenceof perturbationswithkind performedon the Hénonmapping [11] in differentvaluesof a.which smallrandomperturbations~ and~n with a Fig.4 showsthat for thelarger perturbationsthegaussiandistributionwere addedto theiterationfor- fluctuationsbecomesolargethat sectionswithinmulae: two pairs fuse.Thechangevisible in fig. 5 is even

2 moredramatic asfor the largerperturbationsthe= 1 +y,~—ax~+~, Yn+1 =bx~+flfl• (1) orbitremainsontheinitialsix-arcattractorfor

Thetestswere performedfor b = 0.3 like in all inves- very few iterationsonly andthenjumpsontothefour-tigationson the Hénonmapping,andfor two values arc attractorknown from thepreviousfigure.of a: a = 1.4, for whichHénonhas foundanattractor In the presentedteststhemicroscopicrandomwith theCantorsetstructureconfirmedlaterby more perturbations(theyare reallymicroscopicin corn-rigorousmethods[12—14],andfora = 1.08,for parisonwith thesizeof thewhole attractor)gener-which Curry [12] hasfoundtwo coexistingattractors, atedmicroscopicfluctuationsonly in thecaseofeachcomposedof few separatesections,andexhibit- theHénon attractorbut in the caseof the Curryat-ing thenoisyperiodicity analogousto that reported tractorssimilarperturbationscausedfluctuationsby Lorenz [15] for a map of aninterval, visible on amacroscopicscale.Thisdifferencecanbe

probablyexplainedin the following way. Perturba-* Supportedin partby thePolishMinistry of Science,Tech- tionstransversalto theunstablemanifoldof the hy-

nologyandHigher Education,ProjectM.RJ.7. perbolicfixed pointare contractedveryrapidly dur-

0 031.9163/82/0000—0000/$02.75© 1982 North-Holland 315

Volume92A,number7 PHYSICSLETTERS 29 November1982

a) ______________ c)______________

0.4 0.4 0.4

0.3 ~ 0.3 ~ 0.3

10 05 005 10 15 15 10~0 ~ 15 15 10050 05~ ~

~z0.00005 arQ.0005

Fig. 1. TheHénonattractor:(a)without fluctuations,(b) and(c) in presenceof fluctuations.The initial point markedwithanasterisk.

a) b) c)0.2~ 0.2~

020 ~ 020 ~ 020 j’..

~ ,~-..

..--. ~ _a~___ ~ —

:: ~ :: ~

— ~- -‘017 . ~.. 017 ‘.. .~ 017 ~ “I..

0.16 •‘~‘~~‘- 0.16 -. “-P- 0,16

0.15 ‘“‘ ____________________________ 0.15 I 0.150.55 0.60 0.65 0.70 0.55 0.60 0.65 0.70 0.55 0.60 0.65 0.70

d’~0.00005 o~0.0005

Fig. 2. Enlargementof thesmall squaresfrom fig. 1.

a) ~) c)0.191F _____~______‘~~~ ——~ 0.19~p-~ 0.19. •,, . .

0190L~ ~ oi~4 0190

U

0189k- ~. 0189 0189

0188 0168 .., 0188

0187 0187 0187

0.186 0.186 ‘: , 0.166

0.18E L........,._.__L- 0.18~ 0.16!0.625 0.630 0.635 0.640 0.625 0.630 0.635 0.640 0.625 0.630 0.635 0.640

~0.00005 0.0005

Fig. 3. Enlargementof the smallsquaresfrom fig. 2.

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Volume92A, number7 PHYSICSLETTERS 29 November1982

a) b) c)

0.4 04 0.4

0.3 ~ 0.3 0.3

0.2 0.2 0,2

0.1 0.1 0.1

0 0 0

-0,1 J -0.1 ,J -0,1

-0.2 -0.2 ,..1’ -0,2

-0.3 -0.3 -0.3

-0.4 -0.4 -0.4

-1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 -15 -1.0 -0.5 0 0.5 1.0 1.5

= 0.0001 0.001

Fig. 4. Picturesobtainedfor theinitial point closeto thefour-arcattractorreportedby Curry [12].

ci) b) c)

0.2 0.2~ .,~ 0.2

-~ ) ) .:~‘ U

-0.2 -02 -0.2

-0.3 — -0.3 — -0.3

-0,4 -0.4 -Oh.........L ....... .L.... J~..J... I ——

-1.5-1.0 -0.5 0 0.5 101.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.5 -00 -0.5 0 0.5 1.0 1.5

0.0001 0.001

Fig. 5. Thesameasin fig. 4 but for theinitial point closeto thesix-arcattractorof Curry.

ing succeedingiterations,andthe transversalfluctua- orbitsin unperturbedsystemsare denseonthelions are practically of theorderof magnitudeof a whole unstablemanifolds.However,in the caseofsingleperturbation.On thecontrary,perturbations the noisyperiodicmotionsthegeneratedfluctuationsalongtheunstablemanifold cansummarizeandthe canbemacroscopic:the motionscanbecomemoreresultingfluctuationscanbemuchlarger.However, noisyor eventheir orbitscanbe switchedfrom onetheselongitudinal fluctuationscanbe noticedonly noisy-periodicattractorto another.It suggeststhatif the attractoris notextendedon thewhole unstable the noisy periodicmotionsare especiallysensitivemanifold. Onehasjust sucha situationin the caseof to smallrandomperturbations.theCurry attractorsbutnot in thecaseof the Hénon Theaboveconclusionneedsaverificationbothbyattractor[13]. Leavingthe questionasto whether investigationson othersystems,andby applicationthe aboveexplanationis correctfor a morecomplete of morerigorousmathematicalmethodsto the testedstudy,wecansummarizethe presentedresultsas cases.Oneof the problemsarisingfor thelattermatterfollows. Smallmicroscopicperturbationsdo not is connectedwith thefact that theperturbationsusedchangeon themacroscopicscalethe motionswhich in thepresentedtestsareunboundedin principle(inare “completely” chaoticin the sensethat their practicea computercangenerate,at least for certain

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Volume92A,number7 PHYSICSLETTERS 29 November1982

seeds,singleperturbationswhosevalue ismany orders [3] S.T. Dembifiski, A. KossakowskiandL. Wolniewicz,

largerthana, sothatsuchasingleperturbationis no Z. Phys.827 (1977) 281.longersmall), andit is unclearwhetherthedefinition [4] R.B. SchaeferandC.R.Willis, Phys.Rev.A13 (1976)

1874.of anattractorin the presenceof smallrandomper- [5] A. Zardecki,Phys.Rev. A23 (1980) 1281.

turbationsgiven by Ruelle [16] isapplicable,from [6] K. Stefañski,Z. Phys.B45 (1982)351.therigorousstandpoint,to the objectscalledattrac- [7] G. Mayer-KressandH. Haken,Phys.Lett. 82A (1981)

tors in thepresentreport.The problemisnot a trivial 151.one,andlike otherspointedout,will needseparate [8] B. Shraiman,C.E.WayneandP.C.Martin, Phys.Rev.

studies. Lett. 46 (1981)935.[9] J.P.Eckman,Rev.Mod. Phys.53 (1981)643.[10] A. Zardecki,Phys.Lett. 90A (1982) 274.

The authorisindebtedto ProfessorS.T.Dembiñski [11] M. Hénon,Commun.Math.Phys.50 (1976)69.

for inspiringdiscussionsandremarks. [12] J.H.Curry,Commun.Math.Phys.68 (1979) 129.[13] V. FranceschiniandL. Russo,preprint(1980).[14] M. Misiurewicz andB. Szewc,Commun.Math.Phys.75

References (1980)285.[15] EN. Lorenz,in: Nonlineardynamics,Ann.NY Acad.

[1] A.D. Ventceland MJ. Freydlin, Fluctuationsin dynam- Sci. 357 (1980) 282.icalsystemsunderinfluenceof smallrandomperturba- [16] D. Ruelle,preprint IHES/P/81/23(1981).tions(Nauka,Moscow,1979)(in Russian).

[2] S.T. DembaiskiandA. Kossakowski,Z.Phys.B24(1976) 141.

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