Oscillatory Chemical Reaction in a CSTR with Feedback Contro(4)

J.Phys.Chem.A,Vol.101,No.28,19975153

Figure7.Bifurcationdiagrams,simulations;Dashedline,Hopfbifurcationline;areaconfinedbysolidline,regionofoscillations.Darkgrayshadedareawithnumbers,burstingoscillations;graylevelsandnumbersindicatenumberofoscillationsperburst.White,stablesteadystatewithhigh(HI)andlow(LI)iodideconcentration.Black,chaoticoscillations.OSC,domainofperiod-oneoscillations.(a)SetupA,(b)setupB,(c)setup

C.

Figure8.Bifurcationdiagrams,simulations.Parameterspace:(a)logk01-logITforsetupBand(b)logkmax,1-logITforsetupC.LinesandshadedareaasinFigure7.

HIsteadystate.DuringtheapproachtotheHIstate,theiodideconcentrationinthereactorexceedsIT,whichtriggersadecreaseoftheflowrate,depictedinFigure9abythearrowmarkedA-.Torestoretheoscillations,theflowratemustfall

below

5154J.Phys.Chem.A,Vol.101,No.28,1997Figure9.BifurcationdiagramsofCDIreactioninaCSTR.(a)Two-parameterdiagram:solidline,Hopfbifurcationline;dashedline,saddle-nodebifurcations.(b)One-parameter(solution)diagram:[I-]o)3.8×10-4M;solidline,stablesteadystate;dashedline,unstablesteadystate.LI,lowiodidesteadystate;HI,highiodidesteadystate;SN,saddle-nodebifurcationpoint;HB,Hopfbifurcationpoint;US,unstablesteadystate;OSC,period-oneoscillations.Forothersymbols,seetext.

thevalueatwhichtheHIsteadystatebecomesunstable(leftmostdashedlineinFigure9a).Inthisway,burstingisproduced,withadecreaseoftheflowrateassociatedwiththequiescentperiodandanincreasewiththeoscillatoryperiod.

Thefixedratior)[I-]0/[ClO2]0insetupArequiresthattheflowratemustundergolargeamplitudeoscillationsforthesystemtoexhibitbursting.Ontheotherhand,thetotalflowrateandtheratiorchangesimultaneouslyinsetupB,asshowninFigure9abythearrowsmarkedB+andB-.ThusburstinginsetupBrequiresmuchsmallerchangesintheflowrate.ThepathdenotedinFigure9casC+andC-representschangesinsetupCduringburstingoscillations.BurstinginsetupCemergesfromadynamicchangeoftheratior,whilethetotalflowrateremainsalmostunchanged.TheratiorinsetupCchangesonaverageabouttwiceasfastasinsetupB.Therefore,weregardsetupCasourmosteffectivesetuptoexhibitbursting.Unlikebursting,chaosdoesnotseemtobedirectlyconnectedwiththecoexistenceofastablesteadystateandthelimitcycleoscillations.TheonsetofchaosoccurswhenthedynamicflowrateoscillatesintheimmediatevicinityofaHopfbifurcationpoint.Intheuncontrolledsystem,theperiodofoscillationsisonlyslightlyaffectedbytheflowratechangeinsidetheoscillatorydomain,buttheperiodincreasesrapidlywithincreasingflowrateinthevicinityoftheHopfpoint.Fortheonsetofchaosinthesystemwithfeedbackcontrol,itisimportantthattheincreasingperiodaffectspredominantlytheportionofacyclewithhigh[I-]andthustheaverageconcentrationof[I-]alsochangesrapidlywithchangingflowrate.Wesuggestthatthisstrongnonlinearityisamajorreasonfortheonsetofchaosinoursystem.

Dolniketal.

Themodels(eqs3-6)giveresultsingoodagreementwithourexperiments.Simpleperiodicandburstingbehaviorarethemainregimesfoundbothinexperimentsandsimulations.Chaos,asrevealedbysimulations,ispresentonlyinverysmallparametricdomains(seeFigures7and8).Confirmationofchaosinexperimentsrequiresmoredetailedstudy.Weobservedinourexperimentsseveral(upto10)periodsofperiod-twooscillations,butthenoise,suchaspulsesfromtheperistalticpumpsandtemperaturefluctuations,precludedobservationoffurtherstatesoftheperiod-doublingsequence.Bettertemper-aturecontrolandmoreuniformdeliveryofthereactionmixturemightallowexperimentalverificationofthechaoticbehavior.Conclusions

Wehavestudiedthreesimilarfeedbackregulationmecha-nismsinanattempttoexploreinterestingdynamicalbehaviorinasystemwhich,withoutcontrol,displaysonlysteady-stateand/orsimpleperiodicoscillations.Wehaveconfirmedinexperimentsthatburstingemergesasaresultofthefeedbackcontrol.Thecharacteristicsoftheburstingbehaviorsthenumberofspikesperburst,thefrequencyofspiking,andthelengthofthequiescentperiodscaneasilybevariedbyselectingtheappropriatesetupandthetwocontrolparameters:thetargetiodideconcentrationITandthemaximumflowratekmax.ThedynamicalbehavioroftheCDIreactionwithfeedbackcontrolisqualitativelysimilartothedynamicsreportedforaneuronmodel.11,12Burstsofhighfrequencyfiringhavespecialimportanceinbrainfunction.19Wehavedemonstratedasimilarkindofdynamicalactivityforthe“simple”inorganicreactionbetweenchlorinedioxideandiodide,supportingthenotionthatchemicaloscillatorysystemsmayserveasusefuldynamicalmodelsforneurons.

Acknowledgments.WethankDavidBrayforhishelpwiththedataacquisitionandWon-kyuHanandJanaDolnikova,Jr.,fortheirassistancewiththeexperiments.ThisworkwassupportedbytheNationalScienceFoundationandbytheW.M.KeckFoundation.ReferencesandNotes

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