Volume146,number7,8 PHYSICSLETTERSA 11 June1990

MAP DYNAMICS AND NEWTON’S SECONDLAW

PaulE. PHILLIPSON andA.J. ARMSTRONGDepartmentofPhysics,Box390, UniversityofColorado,Boulder, CO80309-0390,USA

Received6 December1989;acceptedfor publication2 April 1990Communicatedby D.D. Hoim

Newton’ssecondlawis examinedforthemotionof aparticlesubjectto a fluctuatingforce appliedimpulsivelyatequalincre-mentsof time. Theresultis coupledmappingswhichgeneralizeprevioustwo-dimensionalmapstudies.Application is madetothemotionof aparticle in asymmetricdoublewell potential.Transitionsbetweenthewells andescapecanbeunderstoodintermsofcomplementarydiffusion andimpulsiveprocesseswhichcanariseby eitherself-generatedorexternallyinducedfluctua-tions.Theresultssuggestthatstochasticdynamicalprocessesmaybe understoodin termsofthemapdynamics.

1. DiscretizationandNewton’ssecondlaw time. In thedoublelimits r—.0, n-+oo theproductn’rbecomesa continuoustimevariableand(1) reduces

The purposeof the presentwork is to investigate to dp/dt=F, recoveringconventionalNewtonianconsequencesof discretizationof Newton’ssecond dynamicsin continuousform.As r departsfrom zerolaw fora singleparticleof massm movinginonedi- thecollisionsbecomelessfrequentbutmoreviolent.mensionx with velocity vaccordingto As a result, for nonlinearforcesinstabilities char-

acteristicof perioddoubling[1], fractal behaviours= T(t)F(x, p, t), of basinboundaries[21 andchaosarisein physical

dt settings.Thesolution of (1) is givenin termsof thefollowing integralequation,

T(t) =lim’r ~ , (1)~-.O k~O I

wherep=mv is theparticlemomentumandF is an p(t) =p(t~)+S dt’ F[x(t’ ), p(t’), t’ ]T(t’)externallyappliedforcewhich mostgenerallycande- tO

pendupon position,momentumandexplicitly upon p(t~)time. This force is assumedto be applied impul- x(t) =x(t0) + —~-— (1—t0)

sivelyat equallyspacedinstantst= ‘r, 2r, 3r Inthe limit ~—+0the particlemoveswith constantye- + ~5dt’ (t—t’ )F[x(t’ ),p(t’), 1’ ]T(l’). (2)locity in time intervalsnr<t< (n+l)r, n=0, 1, ..., m~. At the endsof thesetime intervals the particlesuffersa discontinuouschangeof momentumpro- In theinterval t0= nr< t’ <t= (n+ 1)r only the k= hportionalto rF. Theperiodzservesdynamicallytwo termin T(t’) contributes,sothat the integrandsarefunctions.First, it measuresthe interval between evaluatedat t’ = nt+ ~. In thelimit E—~0oneobtainssuccessiveimpulses.Secondly,it is thecouplingcon- ~ ( I) =o,,+ tF~, F,, F(x,,,p,,, n) , (3a)stantto the impulsiveforceandassuchdictatesthestrengthof each impulse.In the limit r—*0 the col- x(t)=x +~ (t—nt)+lisions becomeinfinitesimalbut also infinitely fre- “ m mquent.Underthisconditiontheparticleis underthe (nt~~~ (n+ 1) ~, (3b)constantsiegeof F whoseeffect is uninterruptedin

0375-9601/90/$03.50 C ElsevierSciencePublishersB.V. (North-Holland) 401

Volume146,number7,8 PHYSICSLETTERSA 11 June1990

wherex,,~x( nr),p,,=—p( nt) aretheparticlepositionandmomentumrespectivelyat t= nr. Eq. (3a) shows x~+~= x~+ (1 —

2tim~+ —~ (J~+ g~), (5b)that sincep is independentoft theparticlemomen-tumsuffersa discontinuoustransitionfrom pntO ~ g~ g( n), g~~

1= M(g~)at t= (n+ I )r, while (3b) shows that the position J~=— dV/dxj~_~,. (Sc)makes a smooth transition in time. Settingt= (n+ l)r at the endof the interval in x(t) results From (Sa) and (5b) the Jacobianof the mappingin [4] is 1—kr. IfA=0 the mappingis areapreserving

and the systemmotion is conservative.If 2>0 the± =p,,+ tF~, (4a) mappingis areacontractingsothat thesystemis dis-

r sipative.At the limiting valueof 2=hr thebalancex,,~1=x,,+ — p,, + — F,, of dissipationagainstthe collision frequencycol-

m mlapsesthe mappingto onedimension.In this limit

(F,,mF(x,,,p,,,n)) . (4b) (Sb) is x,,÷1=x,,+(f,+g,,)/m22and the momen-

Comparisonof (4a) and(4b) indicatesthatT(t) of turn is slavedaccordingto ~ 1= (i~+g,,)/2. Since(1) implies theparticlevelocity satisfiesthe differ- f~,dependsonly upon position it is presumedto beence equation v,,±

1=(x,,+1—x,,)/r, or ~ derivable formally from a potential accordingtom(x,, — x,,— / r. Then(4b) canbeexpressedasF,,= (5c). Finally, g,, functionsas an external force in-m(x,,+1 — 2x,,+x,,_1) /r~.Consistentwith the veloc- dependentof the stateof the particle. Its characterity, the accelerationis givenby a,,~1= (v,,~,— ~ ) ~ is determinedby somepreassignedmappingM ac-This resultsin F,,=ma,,~1which is a statementof cording to (3d). With g,,=0 particularizationsofcasuality:the force actingat time nr mustproduce theseequationshavebeenknown and investigatedits effect of accelerationat the subsequenttime for a longtime.Forexample,iff,, ~ , (5) withg,,= 0(n+ 1) r. The map equations(4) are concededto is formally theHénonattractorregardedasthePoin-predictthe significant dynamicsof the particle.The caresectionfor a three-dimensionalflow [5]. In theremaininginformationof theparticletrajectoryx( t) presentcontextthe Hénonmap couldmodel thedy-duringthetimesbetweenimpulses(duringwhich ki- namicsof a particle in an impulsively applied an-netic energy is conserved)is recoverablefrom the harmonicpotential.Othersystemswhich canbecastcompletesolution (3). Computersolutionsof non- into (5) include the periodicallykicked rotor [6],lineardifferentialequationsnecessarilyinvolve some the motion of protonsin intersectingstorageringsfinite time incrementr which, if sufficiently large, [7], andthe motion of a buckledbeamundergoingcanlead to what Lorenz [3] refersto as “compu- forced lateral vibrations [81. Eqs. (5) representatational chaos”. Discretizationhere is not consid- generalizationof which thesecasesareexamples.Theered to be a requirementfor analysisof systems next sectionwill apply theformulation to a problemmodeled by continuous differential equations. of genericinterestin physicsandchemistry:the mo-Rather, eqs.(4) arepositedtobe themodel itself of tions of a particlein a confinementmodeledby athedynamicsof interest.As a consequencer in any symmetric doublewell potential,within which theapplicationmusthavea physicalorigin: theparticle particlecanmaketransitionsbetweenwells or fromis presumedto bein a collisionalenvironmentwhich which the particlecanescapeentirely.interactswith the particlewith a collision frequencyl/t

The force will be specializedhereto be decom- 2. Map dynamics of a particle confined by aposable into three terms according to F,,= —2p,, symmetricdouble well+f(x,,)+g(n). The term linear in the momentumincorporatesdissipationparametrizedby 2. Eqs. (4) Themotion ofa particleinitially confinedin a re-becomeas a result gion definedby a symmetricdoublewell prototypes

+ = (1 — kr )p,,+ r(f, +g,,) , (5a) systemswhich can exhibit metastability,bistabilityandescape[9]. The particleis presumedsubjectto

402

Volume 146,number7,8 PHYSICSLETTERSA 11 June1990

a force definedby the well andsubjectto impulses,sothat thecauseof transitionwithin the well or es-capefrom the well is noise induced.The dynamicalprocessis traditionally considereda stochasticonewithintheframeworkofsolutionofasuitableFok- . .

ker—Planckor masterequation.We considerit herewithin theframeworkof themapdynamicequations(5). Thereare threefeatureswhich will appearcon- >

cerningnoise inducedtransitionsandescape.First,particletrajectoriescanbetracedin detailwhich aredifficult to depict within the Fokker—Planckap- \proach.Secondly,it will bedemonstratedthatwithin o• I I

theframeworkof classicaldynamicsasencapsulated ~ -i 0.0 I

by (1) transitionscan occur in the absenceof ex- ~/0

ternalnoise:transitionscanbe inducedby self-gen-eratedfluctuationsdueto theonsetofcrisis [2]. Fi- Fig. 1. Doublewell potential V(x)ofeq. (6).Thepotentialmm-nally, impositionof externallyimposedfluctuations imaareatx/a= ±i. Thelocal maximumV

0atx= 0 occursagainleadsto a map formulation which, within a specific atx/a= ~

parameterrange,parallelstheKramerstheory [9,10] 2

for transitionof particlesby diffusion overa poten- x,,+1 =x,, +sy,,+ r(x~—x~)+ g,,tial barrier. For anotherparameterrangethe diffu- masion mechanismgives way to an impulsive mecha- ~y =x,,÷1—x,,, 0~s~1) . (8)nismfor noiseinducedtransitionsmorecloselyalliedto tunneling. The high damping and conservativelimits corre-

The dynamicswill be modeledby the following spondto s= 0 and 1 respectively.Thepresentphys-double well potential with its associatedforce ac- ical picture is completedby assumingthe externalcordingto (5c), driving force is a sourceof noisegiven by

V(x)=V0[(x/a)2—h]2, g,,= (arng)z,, (9)

4V0x,, 2

= — [1 — (x~/a)] (6) whereg isa dimensionlesscouplingconstantandthe

successivemap iteratesz0, z1, ... are computergen-wherea is a rangeparameterand V0 is theheight of eratedrandomnumbersovertheinterval — I ~ z,, t~1.the barrierseparatingthe two wells. The well mm- 1ff,, = 0 this noisesourceinsertedinto (5) resultsinima are at x/a= ±1 andthe rangeof confinement the map dynamicequivalentof the Langevinequa-within awell isapproximately0< x/a I ~ A plot tion describingthe Brownian motion of a free par-of this potential is shown in fig. 1. It is useful to tide [10]. Sincetheequationsarelineartheycanbetransformto dimensionlesscoordinatesx,,/a,y,, and solvedexactly.Assumingthe particlestartsat x0 =0definetwo dimensionlesscontrol parametersr and the result is thatx~—’2Dt(t=nr, n>> 1) wherethes accordingto diffusion constantis D= (ga)

2/422r3.Whena po-

tential ispresentit is moreusefultoconsiderthetra-x,,/a—~x~,y,, =p,,r/ma, jectory x,, of the particlefor successiven. Supposer=4V

0r2/ma2, s= 1 —Ar. (7) theparticlestartsat restandat thebottomoftheright

well (yo=O, x0= 1). Fig. 2 showsthreepossibletra-

Insertionof (6) and (7) into (5) gives the result jectoriesover20000 stepscomputedaccordingto

r2 (8). In fig. 2a g, r, s are such that the particle re-

= sy,,+ r(x,, —x~)+ — g~, mainsconfinedin the right well, executingirregularma motionbecauseof therandomimpulsessuppliedby

403

Volume 146, number7,8 PHYSICSLETTERSA II June1990

2 0 contributeswhat has been referred to as additive

I i ii iii - noise[11] which providesa sourceof inducedfluc-. .. tuations.Additive noiseis fundamentalto stochastic

x odynamicsin physics and chemistry. Self-generated

- I noiseas providing a mechanismfor stochasticpro-

-2 ~— cesseshowever,doesnot fit into that framework. It2 ~b) ariseshere as a necessaryconsequenceof the map

I equations(8).

~. ~ ~ ‘-.. We considerinitially thefirst situationwheng~=0,in which case(8) isatransformedversionof thetwo-

- ._ . I dimensionalcubic map introducedby Holmes [9].-2 Thedynamicsof this mappingwill demonstratethat

2 the causeof transitionor escapedue to self-gener-

I atedfluctuationsis crisis This is mostsimply illus

x0 0 (.~i$ ~~i~/? tratedby the cases=0 reducingthe mappingto onedimension so that v÷1=(r+l)x~—rv~Fig 3ashows iteratesof this map, x versus r, for initialvaluex0= —0.999nearthe bottom of the left well.In the control parameterregion0<r< 1 all iteratesconvergeto the period-oneattractingpoint (x= — 1)

Fig. 2. Solutionofeq. (8),x~versusn,n=0,I NforN=20000 which definesthebottom of the left well. Beginningandx0= I, vo=O. At theseparametervaluestheparticle in the at r= I therecommencesa period-doublingscenarioabsenceof externalnoisewould regressto x= I. (a)g=0.

2: theexternallyimposednoisecausesoscillationof theparticlewhich confinedto the left well which convergesto the pe-remainsin theright well. (b) g=0.25:g is sufficiently largeto nod-building accumulationpoint rac~1.300. Be-induce transitionsbetweenthe wells. (c) g=0.45:g is suffi- yond rac lies a region in which orbits areaperiodic,ciently large that theparticle escapes(the arrowindicatesthe but all iteratesare still confinedto the regionof theparticleescapedto infinity). Thesebehaviours,shownheredue . . . . .left well. Thissituationpersistsuntil the crisis pointto additivenoise,alsooccurwith g=0 dueto self-generatednoiseif randsoresufficientlylarge. r~ 1.5980whenthe attractorsuddenlywidens.The

result is delocalizationof the iteratesover thespacetheexternaldriving force. In fig. 2b theseparameters of bothattractors.Thissituation persiststo theendare suchthat the particleoscillatesin onewell, sub- of the mapping at r~= 2, beyondwhich all orbitssequentlymakingtransitionsbetweenwells at irreg- convergeat infinity. This meansthat if the particleularintervals.Thispictureis in accordwith thatpro- hasan initial valuecloseto the bottom of the rightvided by the stochastic treatmentof escapeand well iteratesof x

0 will convergeto orbits in the righttransitionprocesses[9,10]. Fig. 2c shows that for well provided r<r~.More specifically, ifr<rac= 1.3suitableparametersthe particlecanescapeto infin- theparticlewill oscillatein a periodicorbit confinedity. In this lastsituationthedetailsof thedoublewell in the right well, and if ~~ac 1.3<r< r~=1.598 thearelost andthedynamicsis equivalenttoescapefrom particleis still confinedbutoscillatesin an aperiodica singlewell. As will be discussedelsewhere,escape mannerovermostof this r regionasin fig. 2a. At thefrom a singlewell uncomplicatedby transitionsbe- crisis point i.= r~the iteratesexplodein time overtweenwells can be analyzedfrom (8) by formally bothattractorsregardlessof initial condition. Thereplacingr by — r. resultingdynamicsis similarto fig. 2b,with thepar-

Fluctuationswhich producetransitionor escape tide demonstrating“crisis-induced intermittency”areof two origins. If g,, = 0 the forcef,, canserveas [2]: hoppingbetweenthewells at irregularintervals.a source of self-generatedfluctuations which can In thecontinuumlimit r—*0 (8) reducesto theforcedcausetransitionsin theabsenceofexternalnoise.This double well Duffing equation [12] d

2x/dt2+Adx/providesthe backdropfor the externalforceg,, con- dt+dV/dx=g(t) where V is given by (6). Thisstructedas (9) to behaveasa stochasticforce. It then equationhasbeenstudiedby Ishii et al. [13] for the

404

Volume 146,number7,8 PHYSICSLETTERSA 11 June1990

20- namicsbecomestwo-dimensional.There is in thisS’ 0.0 ~__Jgeneralcasea period-couplingscenariowhich con-

verges at an accumulationpoint given approxi-.0- ‘ matelyby rac_~(ör2—r1 )/(ô—1). d=4.6692...is the

Feigenbaumratio, r1 = 1 +s and r2= (5s2+8s +

s)i/2 (1+s). At r=r~thetwo period-oneorbitsbe-0.0 comeunstable,eachsplittinginto period-two,andat

r= r2 the period-two orbits split into period-four.

Thesebifurcationparameterswerefound usingthe0- ________ procedureof Helleman[7]. Theextrapolationof the

accumulationpoint follows from the procedureusedfor one-dimensionalmappings[14]. Whens> 0 the

-~ 096 2 0 two boundedattractorscorrespondingto theleft and(r) right wells now coexistwith a third attractorof un-

boundedorbitswhich convergeat infinity. If thepar-20- tide startingnear the bottom of the right well iter-

S 0.1 (b) atesto a pointwhich convergesto the left well, it hasmadea transition,but if it iteratesto a point on an

.0- unboundedorbit it will escapeentirely. There is amaximumvalueSma,, suchthat ifs < Smaxtheparticlewill maketransitionssimilar to s=0: well confine-

o.o- mentfor r<r0 andwell transitionsfor r~~<r<r,,~.For5 ~

5max the particle will remainconfinedfor r< r,,,,andescapeif r> r,,,,. The reasonis thatwith increas-

-1.0- ing s the crisis point approachesr,,,, such that at

Smax°.2l2thetwo pointscoalesceat r=r.,,= 1.555.As a consequencethereis no more room for tran-

-2.0- sitionsbetweenwells. Fig. 3b showsa plot of the it-0,96 r’~ 2.0 eratesfor s=0.l. Comparisonwith the s=0 caseof

fig. 3aillustratesthenarrowingoftheregionbetweenFig. 3. Iteratesx versusr generatedaccordingto (8) with g,,= 0. the crisis point and the end of the mapping for(a) s= 0, (b) s= 0.1. For each r value the initial values boundedorbits.x0= —0.999,y~=0wereiterated5000 times(xsm~x,a)subse- Transitionandescapeprocessesare usuallycon-

quentto whichwereplotted thenest4000iterates.Thus,anyin- sideredascausedby an external noisesource,suchterval in theaperiodicregion appearsalmostsolid, while in pe- .

nodicregionsthesepointsareredundantlyperiodicpoints.r~and as (9), which generatesinducedfluctuations, herer arethecrisis point andtheendof themapping respectively, measuredby a non-zerovalueof g. The connectionFor r> r,~all orbitsconvergeatinfinity, betweenthe mechanismsof self-generatedand ex-

ternalnoise is illustratedby studyof thephasespacecaseof a periodicdriving forceg=A cos(wi) which of all initial conditionswhich convergeeitherto theinducestransitionsbetweenthe wells. The orbits left well, the right well or infinity wheng=0. Thedemonstratedby the Duffing equationfeaturingan phasespaceillustratinga representativesituationforexternalperiodicdriving force aresimilarto the or- self-generatednoiseisshownin figs. 4a,4b fors= 0.1.bits generatedby thepresentmap,wherethe control All pointswhich convergeto theleft well areshownparameterr is the counterpartof the strengthA of in red,all pointswhichconvergeto theright well arethe periodicforcein the Duffing equation.Thisim- showninyellow andall pointswhichconvergetoin-plies that this one-dimensionalmapping encapsu- finity areblack. Thebasinboundariesseparatingthelatesmostof thedynamicalfeaturesof theexternally colorsarethe Julia setsfor theseattractors.Fig. 4adriven Duffing equation.Whens> 0 the map dy- for r= 1.5100just shortof thecrisispointshowsthat

405

Volume 146, number7,8 PHYSICSLETTERSA Il June1990

10—

I 1 1 ‘ I ‘ I (a)-2.0 _.J~-1.0 0.0 .0 ~ 2.0

x

to

Y 0

(b)I ‘I I I

-2.0 ~ -1.0 0.0 .0 ,J~ 2.0

x

Fig. 4. Phasespaceof initial conditionswhich illustrate thetopologyof self-generatedfluctuationsfor thesymmetricdoublewell. Theseplotsof y=y~versusx=x

0 showbassinsof attractionfor theleft and right wells. Initial conditionswhich convergeto theleft well areplottedin red (x<0) andinitial conditionswhich convergeto theright well areplotted in yellow (x~>0).Theblack pointsindicateinitial conditionswhichconvergeto infinity. (a) r= 1.51, s=0.l: basinboundariesaresimpleasindicatedby solidcolorsaroundthewellminimaatx= ±1. If theparticleis displacedin oneof thewells it will remainin that well similar to fig. 2a. (b) r=r~=l.5ll2, s=0.1:thisshowsthebreakupof thebasinboundariesinto a salt-and-pepper-likefractal structureattheonsetofcrisis. For anydeparturefromresttheparticlehopsbetweenwells similar to fig. 2b. This illustratesthat crisisis themechanismherefor self-generatednoiseresultingin stochasticbehaviour.Inset:s=0.l, r= 1.6925:Intrusionofblack regionscorrespondingto unboundedorbits. If theparticleiteratestoan(x, y) on an unboundedorbit, it will escapesimilar to fig. 2c.

406

Volume 146, number7,8 PHYSICSLETTERSA ii June1990

to

y 0

—lOt-2.0 ~ -1.0 0.0 1.0 ,/~ 2.0

x

I0

y 0

(b)1 I I

-2.0 ~ -1.0 0.0 .0 ~ 2.0

x

Fig. 5.Topologyof inducedfluctuationsfor thesymmetricdoublewell. Thered,yellowandblackpoints showthebasinboundariesasinfig. 4. The greenpointsarethesuccessiveiteratesof themappings(8) with g,, givenby (9) providinga sourceof noise.Thewhitepointslocatethebottomof eachwell atx= ±1, y=0. (a)Diffusive transitions:(r, s, g)= (0.5,0.5,0.225).Notransitionswould occurifgwerezero.With thisvalue of g it is seenthatthegreenpoints canfilter betweentheredandyellow boundariesseparatingtheleft andrightwells.The passagebackandforth ofthesepointsis indicativeof theprocessof diffusionoverthebarrier. (b) Impulsive transitions:(r,s,g) = (1, 0.5,0.131).Herethegreenpointsjust manageto crossthebasinboundarieslocatedneartheouteredgesofthepotentialwellcloseto x= ±~ Thearrowon theleft indicatesaphasepoint whosepreimagewasin thered region(left well) andlocatedatthisarrowjust inside theyellow regionatx~—,~/iTheright arrow is thenextforward iterationwhichprojectstheparticlecloseto theright wellwith largepositivemomentum.Thesubsequentiteration carriestheparticleinto theregion oftheright well. This illustratesthat if r issufficientlylargetheparticlecanbe impulsivelyprojectedfrom onewell to theother.Inset:Escape:(r, s,g) = (1, 0.5, 0.325),g is nowsolargethat thegreenphasepointsare driven into theblack region.Theparticlecan oscillateuntil g drivesthephasepoint into thisregion.Whenthisoccurstheparticleescapes.

407

Volume 146, number7,8 PHYSICSLETTERSA II June1990

thebasinboundarieshavea simplestructureso that value of r= 1. In this case the green overlay forthe colors in the vicinity of the pointsof stability g= 0.131 (g,,,,~= 0.114)falls short of connectingthex= ±1, y= 0 aresolid.Foranysmallfluctuationaway red and yellow regionat x= 0, yet the particle canfrom thebottom of a well the particlewould remain maketransitions.In thiscasemostgreenpointswhichconfinedtheresimilar to fig. 2a. Fig. 4b showsthe do crossare ableto traversethe yellow—redbound-phasespaceof initial conditionsat the crisis point arieslocatedat the outer edgesof the potentialwellr~=1.5112wherethebassinsofattractionfor thetwo close to x= ±,,J~.In the first case (fig. 5a) accu-wells becomefractal. Theirbreakupgivesthe phase mulation of greenphasepoints over the commonspacea salt-and-pepperappearanceover the entire boundaryat x=0 for r=0.5 indicatesthat the mech-regionof space.This illustratesthat for any initial anismis essentiallydiffusion oventhebarrierwithincondition away from the bottom of eitherwell the the frameworkof the Kramerstheory. In thecontin-behaviourof the particle is essentiallystochasticso uumlimit r—~0with g as a sourceof noise (8) re-that theparticlewould hop betweenwells similarto ducesto differential equationsfor noise driven bi-fig. 2b. The insert in fig. 4b showsthe situationfor stablestochasticflow investigatedby Hanggiet al.r= 1.6925 where the black regions for escapein- [15] within the frameworkof the Fokker—Plancktrude. The black regions would dominate at equation. The transitionsas effected by diffusionr= 1.6945 for which all orbits convergeto infinity, demonstratedhereby the mappings(8) parallelthe

With the additionof the externalforce (9) tran- resultsofthesestudies.As r getslarger,however,cor-sitions and escapecan be inducedfor any r and s respondingto a largerrandillustratedby fig. Sb, dif-valueif g is sufficiently large,as all periodicitiesare fusion givesway to a secondprocessnot within thesmearedout by theexternalnoise [11]. Fig. 5ashows framework of continuous differential equations.the phasespaceplot for s=0.5, r=0.5. In the ab- Specifically,the mechanismof thedynamicschangessenceof externalnoisethe particlewould remainin from diffusive to impulsive. The collisions are rel-one well as in fig. 2a. However,thereexists a g,,,,,, atively harderandthereis relatively a greatertimesuchthattheparticlestarting,say, in theleft well (red betweencollisions.The particlehasenoughkineticregion) could make a transition into the right well energyandsufficienttimebetweenimpulsesto cross(yellow region). Thegreenpointsarethe iterates(x,,, the regionoccupiedby the potentialbarrierbeforev,,, n = 0, 1, 2...) generatedby mapping (8) for the nextcollision. As r increasesthe diffusion mech-(x0=—1, y0=0) andg=0.225 (g~j,,=0.190).These anismgives wayto an impulsive mechanismwhichgreenpointsareanoverlayof theorbit in phasespace is a classicalcounterpartto tunneling. The particlein the presenceof noiseoverthe phasespaceof mi- doesnotpenetrateabarrier,butratherpassesthroughhal conditionsin theabsenceof noise.It is seenthat the regionoccupiedby the barrierwhile the barrierg is sufficient to inducea transitionbetweenwells, is notpresentandprior to its appearanceat thenextas indicatedby the green points which lie on both collision.sidesof the red andyellow border which straddles The dynamicsof transitionsbetweenwells illus-the potentialbarriermaximum at x=0. Eachitera- trated in figs. 5a, 5b persist with increasingg andtion representsa new initial condition for subse- correspondingincreasingfrequencyoverthe param-quenttimeevolution.Whenthis initial conditionco- eterrangeg,,,~<g<g~~~.At umax theextensionoftheincideswith theyellow pointnearestto theredregion, greenpointshasgrown suchthat they obtrudeintothe particlemay crossfrom the left to the right well the black region, correspondingto escape.The insetat thenext iteration.Similarly, oncein theright well, in fig. Sbshowsthis for thecases=0.5,r= 1, g=0.325when subsequentiterations lead to a point which (gmax = 0.135). In this situationthe particlewill os-overlaysa red point the particlemaycrossfrom the cillate until it hits a phasepoint in the black regionright well to the left well. As g is increasedthe den- for the first time. This resultsin subsequentescape,sity of greenpoints which straddlethe red—yellow as shown in fig. 2c. In generalgm,, andgma,, heraldborder increaseswith a correspondingincreaseof the onsetof noise inducedtransitionandescapere-frequencyof barriercrossing.Fig. Sbshowsa similar spectively.As r and s increase,the progressivein-situationfor s = 0.5again,butat thesomewhatlarger trusionof theblack regionsindicatesprogressivere-

408

Volume 146,number7,8 PHYSICSLETTERSA 11 June1990

duction of dissipationand increasingly impulsive —

characterto the dynamics.When r ands are suffi- , tat

ciently largeescapeultimately dominatestransition -

betweenwells. Whensucha pointis reachedthebasinboundariesfor escapealsobecomefractal [16]. Inthis limiting situationall particleorbits convergeat -

infinity for arbitrarily smallg values.The transitionand escaperatespredictedby the

presentmodel are bothsimply exponential.We as-sumefirst thatr, s, g aresuchthatthe particleenters -

the left well at the kth iterate,definedask= 0, andmakesa transitionto the right well at k=n. Due tothe stochasticnatureof thenoise,beit selfgenerated ~

0—~-- I I T’~”—t-----—i-~___.L..._I I

or induced,eachcrossinginitiatesa new trajectory n I x 0

within the well which terminatesat the subsequentcrossing.Thiscrossingagaininitiatesa newhistory, 0.014

and so on. If N( n) denotesthe numberof such n-crossingsin N steps,the probability of the particle -

havingentereda well at k= 0 remainingin the well -

after n subsequentstepsis P(n)=N(n)/NasN—’cx~.Fig. 6a shows a plot of P(n) versusn for (r, s, —

g)=(l.0, 0.1, 0.3) for a history of N=108 itera-

tions.Forn> 10 thedatapointsarefit within 1%byP(n)=[l—exp(—y)]exp(—ny), y=7.2xl03. -

Thereis also a sharppeak for smalln which is seen “ -

on the insetplot of In P(n) versusn. This peakin- - N.dicatesshort-timevirtual transitioneventswhich oc- 0 - I

cur when the particle, hovering near x=0 corre- 0 I x I0~

spondingto thetop of the well, jumpsfrom onewellto the otherandthenjumpsbackagainwithin 5—10 Fig. 6. ComputerstudyofP(n)versusn whereP(n) istheprob-

iterations.Wheng?’gmaxthe probability for escape ability of transition or escapeafter n steps.Both exponential

displaysthesameexponentialdependence.P(n) for curvesfit within 1% by P(n)= [I — exp( — y)] exp(— ny). (a)escapeis the probability that the particle, having Transition: (r, s,g, y) = (1.0,0.1,0.3, 7.2x10 3), Inset:in P(n)startedat k= 0 atthebottom of eitherwell (x

0 = ±1, versusnto showshorttimerapidtransitionsbackandforthwhenparticle hoversnearthetop of thebarrier. (b) Escape:(r, s, g,

Yo’°) hasnot escapedafter n subsequentsteps.An ~ (1.0,0.5,0.35,8.6x l0-~).exampleof escapeis shown in fig. 6b for (r, s,g)=(l.0, 0.5, 0.35) for which y=8.6xl0

3. Theexponentiallydistributedcollisionswhich arisehere be stochasticor not stochastic,diffusive or not dif-are characteristicof stochastictheoriesof chemical fusive, chaoticor not chaoticdependingupon thereaction rates[171. choice of forcesandupon valuesof the controlpa-

The presentdoublewell example(6) of the dis- rameterswhich measurethe strengthof the particlecretized Newtonian equations (5) suggeststhat interactions.chemicaldynamicsmaybe formulatedasa stochas-tic processwithin the frameworkof map dynamics. AcknowledgementMore generally,theseequationscanbe extendedtoinclude systemswhosecollisional environmentare Oneof us, A.J.A., wishesto acknowledgesupportdirectly reflectedby theinherentlydiscretizednature by the U.S. Departmentof Energy,GrantNo. DE-of themap equations.Thedynamicscanbetunedto FGO2-86ER45236.

409

Volume 146, number7,8 PHYSICSLETTERSA 11 June1990

References [9] C.W. Gardner,Handbookofstochasticprocesses(Springer,Berlin, 1983).

[10] H.A. Kramers,Physica7 (1940) 284.[1] HG. Schuster,Deterministicchaos,an introduction, 2nd [II] J.P.Crutchfield, J.D. Farmer andBA. Huberman,Phys.

Ed. (Physik-Verlag,Weinheim,1988). Rep. 92 (1982)45.[21C. Grebogi,E. Ott, F. RomeirasandJ.A. Yorke,Phys.Rev. [121 H.T. Davis, Introduction to nonlinear differential and

A 36 (1987)5365. integralequations(Dover,New York, 1962).[3] E.N. Lorenz,PhysicaD 35(1989)299. [131 H. lshii, H. FujisakaandM. Inoue.Phys.Lett. A 116 (1986)[4] A.J. Lichtenberg and MA. Lieberman, Regular and 257.

stochasticmotion (Springer,Berlin, 1983). [141 P.E.Phillipson,Commun.Math.Phys.111(1987)137.[5] M. Hénon,Commun.Math.Phys.50 (1976) 69. [15] P. I-Ianggi, P. Jung and F. Marchesoni,J. Stat. Phys. 54[6] B.V. Chirikov, Phys.Rep. 52 (1979)463. (1989)1367.[7] R.H.G.Helleman, in: Fundamentalproblemsin statistical [16] Y. YamaguchiandN. Mishima, Phys.Lett. A 109 (1985)

mechanics,Vol. 5, ed. E.G.D. Cohen (North-Holland, 196.Amsterdam,1980) pp. 165—233. [17] B. Gaveau,J.T. Hynes,R. KapralandM. Moreau,J. Stat.

[8]P.Holmes,Trans.R.Soc.A292 (1979)419. Phys.56(1989)895.

410