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1、PERIODTHREEIMPLIESCHAOSTIEN-YIENLIANDJAMESA.YORKE1.Introduction.Thewayphenomenaorprocessesevolveorchangeintimeisoftendescribedbydifferentialequationsordifferenceequations.Oneofthesimplestmathematicalsituationsoccurswhenthephenomenoncanbedescribedbyasinglenumberas,forexampl
2、e,whenthenumberofchildrensusceptibletosomediseaseatthebeginningofaschoolyearcanbeestimatedpurelyasafunctionofthenumberforthepreviousyear.Thatis,whenthenumberX,+Iatthebeginningofthen+1styear(ortimeperiod)canbewrittenwhereFmapsanintervalJintoitself.Ofcoursesuchamodelfortheye
3、arbyyearprogressofthediseasewouldbeverysimplisticandwouldcontainonlyashadowofthemorecomplicatedphenomena.Forotherphenomenathismodelmightbemoreaccurate.Thisequationhasbeenusedsuccessfullytomodelthedistributionofpointsofimpactonaspinningbitforoilwelldrilling,asmentionedin[8,
4、111,knowingthisdistributionishelpfulinpredictingunevenwearofthebit.Foranotherexample,ifapopulationofinsectshasdiscretegenerations,thesizeofthen+1stgenerationwillbeafunctionofthenth.AreasonablemodelwouldthenbeageneralizedlogisticequationArelatedmodelforinsectpopulationswasd
5、iscussedbyUtidain[lo].SeealsoOstereta1[14,15],FIG.1.ForK=1,r=3.9,withx,=.5,theabovegraphisobtainedbyiteratingEq.(1.2)19times.Atrightthe20valuesarerepeatedinsummary.Novalueoccurstwice.Whilex,=.975andx,,=,973areclosetogether,thebehaviorisnotperiodicwithperiod8sincex,,=.222.T
6、hesemodelsarehighlysimplified,yeteventhisapparentlysimpleequation(1.2)mayhavesurprisinglycomplicateddynamicbehavior.SeeFigureI.Weapproachtheseequationswiththeviewpointthatirregularitiesandchaoticoscillationsofcomplicatedphenomenamaysometimesbeunderstoodintermsofthesimplemo
7、del,evenifthatmodelisnotsufficientlysophisticatedtoallowaccuratenumericalpredictions.Lorenz[1-4]tookthispointofviewinstudyingturbulentbehaviorinafascinatingseriesofpapers.Heshowedthatacertaincomplicatedfluidflowcouldbemodelled986TIEN-YIENLIANDJ.A.YORKE[Decemberbysuchaseque
8、ncex,F(x),FZ(x),...,whichretainedsomeofthechaoticaspectsoftheoriginalflow.SeeFigure2.Inth
