the cauchy atlas on the manifold of all complete ode solutions

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1、THECAUCHYATLASONTHEMANIFOLDOFALLCOMPLETEODESOLUTIONSPETRCHLADEK,LUBOM´´IRKLAPKAAbstract.Inthispaperthenecessaryandsufficientconditionsfor1amappingtobethedependenceofthecompletesolutionofsomeCfirst-orderordinarydifferentialequationontheinitialCauchyconditiona

2、rededuced.TheresultisobtainedbystudyingtheCauchyatlasonamanifoldofcompletesolutions.Theproofisconstructive-thecorrespondingdifferentialequationisobtained.Theautonomouscaseandthelinearcasearediscussed.TherelationtotheSincovfunctionalequationisclarified.Math

3、ematicsSubjectClassification(2000).34A12,34C30,37B55.Keywords.First-orderordinarydifferentialequations,dependenceofsolutionsontheCauchyconditions,theCauchyatlas,maximalflow,localone-parametergroup,groupoftransformations,theSincovfunctionalequation.1.Introdu

4、ctionInSection2,weintroducethefirst-orderordinarydifferentialequationandtheCauchyatlasonamanifoldofcompletesolutions.Fournecessarycon-ditionsforamappingtobethecoordinatestransformationoftheCauchyatlasaregiven.InSection3,wealsoprovethesufficientconditions(5)f

5、orasystemF={Fτσ}τ,σ∈RoffunctionstobetheCauchyatlascoordinatestransforma-tions.Thenecessaryconditionsarewell-knownandincludedinmanypapersandmonographs(see,e.g.,[2],[7]),whereasthesufficientconditionsarepre-sentedhere,asfarastheauthorsknow,forthefirsttime.The

6、seconditionsareintheformofthecompositionofiterationsinthenon-autonomousdy-arXiv:math/0312317v1[math.CA]16Dec2003namicalsystemstheory(see,e.g.,[4]).Sections4,5and6containspecialcasesofthefirst-orderordinarydiffer-entialequations.Moreprecisely,wediscussherea

7、utonomousequationsandlinearinhomogeneousequations.Section7containsasimpleexample.2.TheCauchyatlasLetMbeaset,nanaturalnumber,Ianindexset.Acoordinateschartisabijection;itsdomainisasubsetofManditscodomainisasubsetofRn.Asystemofcoordinatescharts{ϕτ}τ∈Iisanat

8、lasifandonlyiftheunionofdomainsofallcoordinateschartsisequaltoM.Amanifoldisa1setendowedwithanatlas.BycoordinatestransformationonamanifoldwemeanthemappingFτρ:ϕρ(Dom(ϕτ)∩Dom(ϕρ))∋a7→−1(1)ϕτ(ϕρ(a))∈ϕτ(Dom(ϕτ)∩Dom(ϕρ)),whereϕτ