计算Ulam映射高阶关联函数的数论方法（英文）-豆柴文库

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计算Ulam映射高阶关联函数的数论方法（英文）
Introduction
TheUlammap,alsoknownasthelogisticmap,isaone-dimensionaldiscretedynamicalsystemthathasbeenusedextensivelyinthestudyofchaostheoryandrelatedfields.Thismapisdefinedas:
x<sub>n+1</sub>=1-a*x<sub>n</sub>^2
wherex<sub>n</sub>isthevalueofthesystemattimen,aisaconstantbetween0and2,andtheinitialvaluex<sub>0</sub>ischosenrandomly.TheUlammapexhibitschaoticbehaviorforcertainvaluesofa,andhasbeenshowntobeusefulinmodelingavarietyofphysical,biological,andsocialphenomena.
Inrecentyears,therehasbeenincreasedinterestinstudyingtheUlammap'shigher-ordercorrelationfunctions,whichdescribethestatisticalpropertiesofthesystemoverlongtimescales.Inthispaper,wewilldescribeanumber-theoreticmethodforcomputingthesecorrelationfunctions,anddemonstrateitseffectivenessusingnumericalsimulations.
Method
TocomputetheUlammap'shigher-ordercorrelationfunctions,wewilluseamethodbasedonnumbertheory.Specifically,wewillmakeuseoftheMöbiusinversionformula,apowerfultoolinnumbertheorythatallowsustoexpresssumsoverdivisorsofanumberintermsofthevaluesofafunctiononthosedivisors.TheMöbiusfunctionμ(n)isdefinedasfollows:
-μ(1)=1
-μ(n)=0ifnhasaprimefactorsquared
-μ(n)=(-1)^kifnistheproductofkdistinctprimes
UsingtheMöbiusfunction,wecanexpressthecorrelationfunctionsoftheUlammapintermsoftheMöbiusfunctionandcertainotherfunctionsofthesystem.
Thecorrelationfunctionsaregivenby:
C<sub>k</sub>=<x<sub>n</sub>x<sub>n+k</sub>>
wherethebracketsdenoteaveragingovertimeandx<sub>n</sub>isthevalueofthesystemattimen.UsingtheMöbiusfunction,wecanexpressC<sub>k</sub>asfollows:
C<sub>k</sub>=1/k*∑<sub>d|k</sub>μ(d)F<sub>k/d</sub>
whereF<sub>n</sub>isafunctionoftheUlammapgivenby:
F<sub>n</sub>=<x<sub>i</sub>x<sub>i+n</sub>>-<x<sub>i</sub>>^2
wherethebracketsdenoteaveragingovertimeandiisarandomlychosenstartingtime.
Thus,tocomputethecorrelationfunctionsoftheUlammap,weneedtocomputethefunctionF<sub>n</sub>foreachvalueofn,andthenusetheMöbiusinversionformulatoexpressthecorrelationfunctionsintermsofthefunctionF<sub>n</sub