Generalized models as a universal approach to the

arXiv:nlin/0601061v1[nlin.CD]29Jan2006GeneralizedmodelsasauniversalapproachtotheanalysisofnonlineardynamicalsystemsThiloGross∗AGNichtlineareDynamik,Universit¨atPotsdam,amNeuenPalais10,14469Potsdam,GermanyUlrikeFeudelICBM,CarlvonOssietzkyUniverit¨at,PF2503,26111Oldenburg,Germany(Dated:February5,2008)Wepresentauniversalapproachtotheinvestigationofthedynamicsingeneralizedmodels.Inthesemodelstheprocessesthataretakenintoaccountarenotrestrictedtospecificfunctionalforms.Thereforeasinglegeneralizedmodelscandescribeaclassofsystemswhichshareasimilarstructure.Despitethisgenerality,theproposedapproachallowsustostudythedynamicalpropertiesofgeneralizedmodelsefficientlyintheframeworkoflocalbifurcationtheory.Theapproachisbasedonanormalizationprocedurethatisusedtoidentifynaturalparametersofthesystem.TheJacobianinasteadystateisthenderivedasafunctionoftheseparameters.Theanalyticalcomputationoflocalbifurcationsusingcomputeralgebrarevealsconditionsforthelocalasymptoticstabilityofsteadystatesandprovidescertaininsightsontheglobaldynamicsofthesystem.Theproposedapproachyieldsacloseconnectionbetweenmodellingandnonlineardynamics.Weillustratetheinvestigationofgeneralizedmodelsbyconsideringexamplesfromthreedifferentdisciplinesofscience:asocio-economicmodelofdynasticcyclesinchina,amodelforacoupledlasersystemandageneralecologicalfoodweb.PACSnumbers:05.45.-aKeywords:GenerlaizedModels,BifurcationAnalysis,Balanceequations,PopulationDynamics,DynasticCycle,CoupledLasersI.INTRODUCTIONDynamicalsystemsareusedtostudyphenomenafromdiversedisciplinesofsciencesuchaslaserphysics,pop-ulationecology,socio-economicstudiesandmanymore.Thecorrespondingmathematicalmodelshaveoftentheformofbalanceequations,inwhichthetimeevolutionofthestatevariablesisdeterminedbygainandlossterms.Dependingontheprocessesthataretakenintoaccount,asinglestatevariablecanbeeffectedbyseveralgainsandlosses.Inthemodelingprocessthemodelerusu-allydecidesfirstwhichprocessesareimportantandneedtobeincludedinthemodel.Theseprocessesdeterminethestructureofthemodel.Inthesecondstepeachofthetermsisdescribedbyaspecificmathematicalfunc-tion,whichcanbebasedontheoreticalreasoningorem-piricalevidence.Inthiswayaspecificmodelforthephenomenonunderconsiderationisconstructed.Theinvestigationofspecificmodelsisapowerfulapproach,thathas,inmanycases,revealedinterestinginsights.However,thespecificmathematicalfunctionsonwhichthesemodelsarebaseddependcriticallyonthemodelersknowledgeofthesystem.Whilethemodelermayhavemuchinformationaboutcertainprocesses,othersmaybeknownorbelievedtoexistwhichareverydifficulttoquantify.Specificmodelsarethereforeoftenbasedonalarge∗[e-mail]wellbyconsideringanexamplefromecology:Theso-calledHollingtype-IIfunctionisregularlyusedtode-scribethepredator-preyinteractioninfoodwebsandfoodchains[1].Thisfunctiontakesmajorbiologicalef-fectsintoaccount.But,itcannotpossiblycaptureallthesubtlebiologicaldetailsthatexistinnature.Suchdetailsmayinvolvetheformationofswarmstoconfusepredators,adaptationofthepreytohighpredatordensi-ties,densitydependentchangesinthepredator’sforagingstrategytonameafew.Inmodelssuchdetailsareof-tenomittedonpurposesincethemodelerisinterestedingeneralinsightsthatdonotdependonspecificproper-tiesofthesystemunderconsideration.Butontheotherhand,certaindetailsmayhaveastrongimpactonthequalitativebehaviorofthesystem.Forinstance,ithasbeenshownthatevenminorvariationsinthefunctionalformofthepredator-preyinteractioncanhaveastrongimpactonthesystem’sstability[2,3].Incontrasttospecificfunctionalforms,thebasicstruc-tureofthesystemismucheasiertodetermine.Forin-stanceinourecologicalexampleitiseasiertosaythatthepredationdependsonthepopulationdensityofpredatorandpreythantoderivetheexactfunctionalformthatquantifiesthisdependence.Itcanthereforebeusefultoconsidergeneralizedmodelswhichdescribethestructureofasystemintermsofgainsandlossesbutdonotrestricttheseprocessestospecificfunctionalforms.Ifasystemisconsideredinwhichsomeprocessesareknownwithahighdegreeofcertaintywhileothersremainuncertainitcanbeusefultostudypartiallygeneralizedmodelsinwhichsomeprocessesaredescribedbyspecificfunctional2forms,whileothersremaingeneral.Inothercasesfullygeneralizedmodelsinwhichallprocessesaremodeledbygeneralizedfunctionscanbeadvantageous.Generalizedmodelsdescribesystemswithahighergeneralitythanspecificmodels.Theydependonlessassumptionsandenabletheresearchertoconsiderthesystemfromanab-stractpointofview.Despitethegeneralityofgeneralizedmodels,itispos-sibletocomputethestabilityofasteadystateinthesemodelsanalytically.Thisrevealstheexactrelationshipbetweenthequalitativefeaturesoffunctionalformsandthelocalbifurcationsofsteadystates[2].Inthiswaytheapplicationofgeneralizedmodelsenablesustoinvesti-gatethelocalstabilityofsystemswithahighdegreeofgenerality.Moreover,wecandrawcertainconclusionsontheglobaldynamicsofthesystemunderconsideration.Forinstance,thepresenceofchaoticorquasiperiodicdy-namicsaswellastheexistenceofhomoclinicbifurcationscanbededucedformcertainlocalbifurcationsofhighercodimensio