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1、PhysicaD149(2001)174–196FunctionaldynamicsII:SyntacticstructureNaotoKataokaa;,KunihikoKanekobaResearchInstituteforElectricalScience,HokkaidoUniversity,N12W6Sapporo060-0812,JapanbDepartmentofPureandAppliedSciences,UniversityofTokyo,Komaba,Meguro-ku,Tokyo153,JapanReceive
2、d29June1999;receivedinrevisedform31May2000;accepted16November2000CommunicatedbyY.KuramotoAbstractFunctionaldynamics,introducedinapreviouspaper[PhysicaD138(2000)225–250]isanalyzed,focusingontheformationofahierarchicalruletodeterminethedynamicsofthefunctionalvalue.Tostudy
3、theperiodic(ornon-fixed)solution,thefunctionaldynamicsisseparatedintofixedandnon-fixedparts.Itisshownthatthefixedpartsgenerateaone-dimensionalmapbywhichthedynamicsofthefunctionalvaluesofsomeotherpartsaredetermined.Piecewiselinearmapswithmultiplebranchesaregenerallycreated,w
4、hileanarbitraryone-dimensionalmapcanbeembeddedintothisfunctionaldynamicsiftheinitialfunctioncoincideswiththeidentityfunctionoverafiniteinterval.Next,thedynamicsdeterminedbytheone-dimensionalmapcanagaingeneratea`meta-map',whichdeterminesthedynamicsofthegeneratedmap.Thishi
5、erarchyofmeta-rulescancontinuerecursively.Itisalsoshownthatthedynamicscanproduce`meta-chaos'withanorbitalinstabilitythatisstrongerthanexponential.Therelevanceofthegeneratedhierarchytobiologicalandlanguagesystemsisdiscussed,inrelationwiththeformationofsyntaxofanetwork.©2
6、001ElsevierScienceB.V.Allrightsreserved.Keywords:Functionaldynamics;Syntacticstructure;Iteratedmap;Self-reference1.IntroductionInapreviouspaper[1](tobereferredasI),weintroducedfunctionaldynamicstoinvestigatethearticulationprocesscarriedoutonaninitiallyinarticulatenetwor
7、k.Inthefunctionaldynamics,objectsandrulesarenotseparatedinthebeginning,andwestudyhowobjectsandrulesappearfromaninarticulatenetworkthroughiterationsoffunctions.Thefunctionaldynamicsprovidesasimpleuniversalmodelforthisappearance.InthegeneralintroductionofthepaperI,wediscu
8、ssfiverequisitesforabiologicalsystem,whiletwoofthemareexplicitlystudied.Thesearethefollowing:Inseparabilityofr
