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1、Ecientlyandeasilyintegratingdi erentialequationswithJiTCODE,JiTCDDE,andJiTCSDEGerritAnsmann1,2,31)DepartmentofEpileptology,UniversityofBonn,Sigmund-Freud-Strae25,53105Bonn,Germany2)HelmholtzInstituteforRadiationandNuclearPhysics,UniversityofBonn,Nussallee14{16,53115Bonn,Germany3)Interdiscip
2、linaryCenterforComplexSystems,UniversityofBonn,Br?uhlerStrae7,53175Bonn,GermanyWepresentafamilyofPythonmodulesforthenumericalintegrationofordinary,delay,orstochasticdif-ferentialequations.Thekeyfeaturesarethattheuserentersthederivativesymbolicallyanditisjust-in-time-compiled,allowingtheusert
3、oecientlyintegratedi erentialequationsfromahigher-levelinterpretedlanguage.Thepresentedmodulesareparticularlysuitedforlargesystemsofdi erentialequationssuchasusedtodescribedynamicsoncomplexnetworks.Throughtheselectedmethodofinput,thepresentedmodulesalsoallowtoalmostcompletelyautomatizethepro
4、cessofestimatingregularaswellastransversalLyapunovexponentsforordinaryanddelaydi erentialequations.Weconceptuallydiscussthemodules'design,analyzetheirperformance,anddemonstratetheircapabilitiesbyapplicationtotimelyproblems.Solvingdi erentialequationsisanintegralpartalgorithmsandtools.)Inmoste
5、xistingperformantsoft-ofmanysimulationstudiesinvariousscienti cwaredesignsandwork
ows,theuserspeci esthederiva- eldsrangingfromphysicsoverneurosciencetotiveinalower-levellanguage,andthentheyeitherhaveeconomicsaswellasofthetheoreticalinvestiga-tocompileitthemselvesoritiscompiledontheirbehalfti
6、onsofdynamicalsystems.Asthistaskcanoftenbythesoftware(seee.g.,Ref.9).Inparticularwhenwork-onlybesolvednumerically,thereisawideneedingwithahigher-levelprogramminglanguage,thisre-forsoftwarededicatedtoit.Tworelevantbutof-quirestheusertoswitchtoanotherlanguage.Moreover,tencon
ictingcriteriaforth
7、edesignofsuchaitisunfeasibletoexplicitlyspecifylargesystemsofdif-softwareareusabilityandeciency.Thelatterisferentialequations,whichinparticularoccurduringtheofparticularinterestfortheintegrationoflargesimulationofdynamicsoncomplexnetworks10,
