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Volume79A, number4 PHYSICSLETFERS 13 October1980
OCCURRENCE OF STRANGE ATTRACTORS IN THREE-DIMENSIONAL VOLTERRA EQUATIONS
A. ARNEODOLaboratoiredePhysiqueThéorique,UniversitédeNice, 06034NiceCedex,France
1
and
P. COULLET andC. TRESSEREquipedeMéchaniqueStatistique,UniversitédeNice, 06034Nice Cedex,France2
Received2 July 1980
A criterion is givenwhichallowsone to constructone-parameterfamilies ofVolterraequationsdisplayingnumericallychaoticbehavior.
In this letter, we give a simplecriterionwhich a1- ~ = N~F1( {N1}) , i, / = 1, ..., n . (1)
lows oneto constructone-parameterfamiliesofVolterraequationsdisplayingstrangeattractors* ~. The possibleexistenceof strangeattractorsin suchA typical exampleof sucha systemis studiednumeri- systemshasbeendiscussedby Smalein ref. [11]. Thecally. Let us recallthat theseequationshavebeenin- particularchoice F1 = — ~
2jv~N1corresponds,in
troducedby Lotka [3] andVolterra [4] in the theory the terminologyof ref. [12] , to the generalizedof biological populations.Furthermoretheyarisenat- Volterraequationsurally in a variety of problemsof modecoupling in dif-ferentbranchesof physicslike laserphysics[5], plasma a V. = N ( . — N = 1 n 2physics[6] andconvectiveinstabilities[7,8]. They ~ i~7i ~ 1
alsoconstitutethenormal form in theunfolding ofsomesingularitiesof vectorfields [9] - Froma physi- As (1), theseequationspresenttheparticularitythatcist’s pointof view, this stifi correspondsto coupling all thehyperplanessomemodescloseto their linearinstability e.g.station- {N } = 0 / EJ .~ ~ {l , n}ary modesin thepresenceof somesymmetriesor os- / ‘ ‘
cilation modeswithin anhypothesisof weakreso- areglobally invariant. Theypossessat mostT2’~ station-nance[10]. arysolutionsaccordingto the variousways to set the
Volterraequationsariseasa particularcaseof the right-memberof (2) equalto zero.GenerallytheN,’sequations representpositivequantitiesandoneis interestedin
thesolutionswhich lie in thepositivephasespace(Q+~fl
~ EquippedeRechercheAssociéeauCNRS. ‘~ ‘2 LaboratoiredePhysiquede laMatièreCondenséeAssocié Thepossibledynamicalbehaviorsin thecasen
auCNRS. havebeen completelyclassified[13] - A particularity* 1 In ref. [1] , Newhouseindicatesthat numericallyobserved of thebidimensionalsystemis its hamiltonianchar-
strangeattractorsmightbefreaksandcould correspondto acterwhenit generatesperiodicorbits [14] . Thischar-theexistenceof periodicsinkswith very long periods.This . .
objectionmightberemovedwhenthereexistsa “strange acteristicdisappearsif one considersthe casen = 3.attractor”arbitrarily closeto a homodlinicorbit F0 asde- Thenonecanobtain limit cyclesissuedfrom Hopffined below.We will ignoresuchobjectionsin this letter [2]. bifurcations[15] - In thecaseof higherdimensionality,
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Volume79A, number4 PHYSICSLETTERS 13 October1980
quasiperiodicbehaviorcanbe envisaged.However,assoonasn = 3, strangeattractorsmaybe displayednumerically.Theaim of this letteris to give a way tofind at leastsome of them.
A fundamentalstepin the investigationof stochas- 1B~ticity in dynamicalsystemsconsistsin proving theex-
xistenceof infinitely manyperiodicorbits [161. Inthis orderfor somethree-dimensionaldifferentialequations,it is enoughto satisfytheconditionsof atheoremby Shil’nikov [171.This theoremstatesthatif, aftersomechangesof coordinates,thedifferential (a)equationis conjugateto
r~=(p+iw)u, ux+iy, r
~Xz, (3)in someneighborhoodof the origin~2, with
—~ > 0, (4) ~x
andif thereexistsan orbit F0 which leavesthe originand returnsto it as t —~+co, then everyneighborhoodof F0 containsa denumerablesetof unstableperiodic Azsolutionsof saddletype (for a preciseformulation see (b)refs. [17—19]).
A heuristicway to constructsystemswherethis Fig. 1. A mechanismto constructthehomoclinic orbit i~
theoremapplies,is to considera one-parameter.t. definedin the main text. (a) Thescrewrepresentspart of amotion towardsthesaddlefocusA on its stablemanifold. The
family of differentialequationssuchthat: unstablemanifoldof A convergestowardsthe stablelimit(i) for ji greaterthansome 1.10 thereexistsa fixed cycleissuedfrom B by Hopf bifurcation. (b) Thehomocinic
point A which is a saddlefocusandsatisfies(4); orbit F0 throughthesaddlefocusA.
(ii) for /1 = I-1H >/.10,a supercriticalHopfbifurca-
tion [20] occursat a distinct fixed pointB andgen-eratesa stableperiodicorbit forp ~ ~ + + ~ = ~, ij =f~(x),
(iii) for ~ > PH~theunstablemanifold of the sad-die focusconvergesto this periodicorbit when it is orstable(fig. la), or the otherattractorsissuedfrom it - -
via bifurcations; X = )‘, ) = Z, Z = —y f3z + fM(x) . (5)(iv) whenp still increases,the size of the attractor Let usnote that for somef,~(x)piecewiselinear we
grows faster than thedistancefrom A to B. havealreadyprovedtheexistenceof a homoclinic or-Thenwe canhopethat for p largeenoughthe unstable bit F
0 [19] , giving convincingargumentsfor therele-manifold of A comesbackcloseenoughto this point vanceof Shil’nikov’s theoremfor thestudyof a wideso that it becomespartof its stablemanifold generat- classof dynamicalsystemsexhibitingchaoticbehavioring thehomoclinic orbit F0 of Shil’nikov’s theorem [23,24].(fig. I b). As far aswe are concernedwith three-dimensional
Indeedthe structureof severalpreviouslydis- Volterraequations,the invarianceof thecoordinatecoveredstrangeattractorscanbe understoodusing this planesN1 = 0, i = 1, 2, 3, preventsusfrom construct-idea. Thiswill be developedin a forthcomingpaper ing ahomoclinic orbit F0 using theabovementioned[21] aroundthestudyof the classof forcedoscilla- heuristicmethod.However,somesmallperturbationtors introducedin ref. [22] of theseequationsfor adequatevaluesof the param-
*2 Using Poincaré’sterminologythe origin is thena saddle etersallows usto get sucha homoclinic orbit. Thus,focus.
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Volume79A, number4 PHYSICSLETFERS 13 October1980
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Volume79A, number4 PHYSICSLETTERS 13 October1980
via argumentsstatedin ref. [18], we can reasonably mapping [26] (fig. 3a,b).For largervaluesof p, thisexpectto get infinitely manyunstableperiodicorbits attractorevolvesuntil (figs. 2e and3c)we comeclose.of saddletype in three-dimensionalVolterraequations to a situationwherea homoclinic orbit F~existstoo. which is biasymptoticto the centralpoint [19] (if
Actually we haveobservednumericallyseveral F~doesexist,the system(6) verifies the conditionsstrangeattractorswith theseequations.Theyhave of Shil’nikov’s theoremwith t replacedby —t [19]).beengeneratedusingtwo successivenumericalpro- Let usnote that theaboveheuristicconstructiondoescedures.ChoosingN1 = N2 = N3 = 1 asthe central not allow thepredictionof sucha homoclinicorbit.equilibriumpoint (without severelossof generality Forstill largervaluesof p, the shapeof theattractorisfor our problem),(2) reads: simpler(fig. 20. This resultsfrom theexistencefor
3 ~het 1.708of a heteroclinic orbit (fig. 4). Despite— the fact that thisheteroclinic orbit goesthroughtheN~•~ c111(l — N1) (6) saddlefocus, thedynamicsmust be regularin its neigh-
borhoodasit canbe shown using methodsinspiredby
with nine independentparametersct~.In the first pro- ref. [27] ; this is contraryto whathappensin the casegram,startingwith eight coefficientschoseneitherat of a homoclinic orbit. However,for p ~ we arerandomor amongpreassignedvalues,we fix the ninth closeenoughto a homoclinic situationwhich inducesin order to havea Hopfbifurcationat the central complicateddynamics.point; thenwe retainthoseconfigurationswherethe We havegenerateddifferent kindsof heteroclinicequilibrium point in oneof thecoordinateplanesis connectionsthrougha saddlefocus,which corresponda saddlefocussatisfying(4). In the secondprogram, to different shapesfor thenumerically displayedwe takethis ninth parameterasa free parameterp, strangeattractor.We havealso observedstrangeat-vary it abovethe Hopfbifurcationvalueandobserve tractorswhen theHopf bifurcationis subcritical [20]the evolutionin p of the dynamicalbehaviorassociat- All theseresultswill be detailedin a forthcomingpapered with (6) [28].
Fig. 2 illustratesa typical evolutionwith Let usconcludethis studyof three-dimensional
0 5 0 ~ 0 1 Volterraequationsby the following remarks:— the additionof nonlineartermsin thefunctionsF~,.
(a11)~= —0.5 —0.1 0.1 . (7) doesnot modify the previousanalysisexceptthat
p 0 1 0 1 theremayexist manyequilibrium solutionsin (R~)3
and consequentlynewconfigurationsleadingto chaos:After a sequenceof subharmonicbifurcations[22,25] — whenall the coefficients are positive,we cannot(figs. 2a,b,c),we reacha strangeattractor(fig. 2d) expectto build strangeattractorswith themechanismwhichmay beconsideredasa suspensionof Henon’s presentedin thisletter sinceno focuscanexistin the
V/ N
xFig. 3. (a)Plot of Henon’smappingH~b(x,y) = (1 ax2 +y, bx), fore = 1.7 andb = —0.001. (b) (c)) A Poincarémapon theplanez = 1 correspondingto theattractorof fig. 2d (fig. 2e).
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Volume79A, number4 PHYSICSLETTERS 13 October1980
[5] W.E. Lamb,Phys.Rev. 134 (1964) 1429.[6] G. Lava!andR. Pellat,in: Plasmaphysics,lectures
deliveredat theSummerSchoolofTheoreticalPhysics(lesHouches,1972)(GordonandBreach,1975).
[71A.C. NewellandJ.A.Whitehead,J. Fluid Mech. 38(1969) 279.
A[8] F.H. Busse,Rep.Prog.Phys.41(1978)1929.
1~ B “ [9] W.F. Lanford, A. Arneodo,P. Coullet,C. Tresserand
—~ J. Coste,Phys.Lett. 78A (1978) 11.x[101 P. Coullet,Thesis,Nice Univ. (1980),unpublished.[111 S. Smale, J. Math.Biol. 3 (1976)5.[121 E.W. Montroll, Proc. IUPAP, Conf. on Stat. Mech. Rice
St.,eds.K. FreedandJ. Light (ChicagoU.P.,Chicago,1972).
[131 L.C. Li, unpublishednotes,Cornell Univ. (1979).Z A [14] J. Guckenheimer,On a codimensiontwo bifurcation,
Univ. ofCalifornia, Santa-Cruzpreprint(1979).Fig. 4. For (ajj)~givenby (7) and~z 1.708,onegetsa [15] J. Coste,J. PeyraudandP. Coullet,SIAM, J. App!.heterodinicconnectionthroughthesaddlefocusand two Math. 36 (1979) 516.distinctequilibriumpointsA andA” lying respectivelyin the [161 S. Newhouse,Talkgivenat theIntern Conf. on Nonlineary = 0 plane andon thex-axis. dynamics(New York, December1979).
[17] L.P. Shil’nikov, Soy.Math.Dokl. 6 (1965)163.
[18] L.P. Shil’nikov, Math. USSRSbornik 10 (1970) 91.coordinateplanes.We arenow investigatingfour-di- [19] A. Arneodo,P.Coullet andC. Tresser,A forcedoscil-
mensionalsystemswherethis possibilityremainsopen. lator with chaoticbehavior:anillustrationof a theoremIn ref. [28] we wifi usethe possibleexistenceof by Shil’nikov, Nice preprintNTH 80/7 (1980), submit-
strangeattractorsin Volterraequationsto describea tedto J. Phys.A.
way to generatestochasticityin a systemof threecou- [20] P. Coullet,C. TresserandA. Arneodo,Proc. Intern.Workshopon Intrinsic stochasticityin plasmas(Cargese,pledoscillators. 1979),eds.G. Lava! andD. Gresillon(LesEditionsde
Physique,Orsay, 1979).
Wewould like to thankProfessorl.A. Sakmarfor a [21] A. Arneodo,P. CoulletandC. Tresser,in preparation.
carefulreadingof the manuscript. [22] P. Coullet,C.TresserandA. Arneodo,Phys.Lett. 72A(1979) 268.
[231O.E.Rossler,Phys.Lett. 57A (1976) 397.References [24] A.L. FabrikantandM.I. Rabinovich,Soy.Phys.JETP
50(1979)311.[1) S. Newhouse,PubL Math. IHES50(1980)101. [25] C. TresserandP. Coullet,C.R.Acad.Sci. 287A (1978)[2] A. Arneodo,P. CoulletandC. Tresser,A new strangeat- 577.
tractor,Nice preprintNTH 80/12(1980), submittedto [26] M. Henon,Commun.Math. Phys.50 (1976)69J. Phys.A. [27] L.P. Shil’nikov, Math.USSRSbornik 6 (1968)427.
[31A.J. Lotka, Proc.Nat. Acad. Sci. 6 (1920)410. [281 A. Arneodo,P. CoulletandC. Tresser,in preparation.[41A. Volterra, Leçonssurle théoriemathématiquede la
lutte pourIa vie (GauthierVillars, Paris, 1931).
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