Bifurcation in economic grawth model to transformed ode

Abstract

Our workfocusesonstudyingtwopapersdeeply,wherewescrutinizeanddis- cuss themainpaperofGuerrinietal.[?], whichisconsideredageneralization of KrawiecandSzydlowskiwork[?]. Whither,theauthorsinvestigatedthedy- namics ofthemodi_edKaldormodelandconcludedgivingthemtheresultthat the limitcyclebehaviorisindependentoftheassumptionthattheinvestment function iss-shaped.Inthisstudy,asanassumption,supposedthelinearfunc- tion I(Y ), andonlythetime-delayparameterplayacrucialroleincreatingthe limit cycle.Also,theyshowedthatforasmalltime-delayparameter,inthelin- ear approximation,theKaldor_KaleckimodelassumestheformoftheLiénard equation. Alinearstabilityanalysisincludingthetimedelayisshowntobean accurate predictorofthecritical T for the_rstbifurcation.As T is increased, the systembifurcatestothelimitcyclebehavior,thentomultipleperiodicand aperiodiccyclesoreventuallytendstowardsthechaoticbehaviorviatheperiod doubling cascaderoutetotheturbulence. Guerrini etalstudiedapossiblebifurcationexpectedtoachangeofthepa- rameter valuesoftheKaldor-Kaleckigrowthmodel,which,theyconsidertwo simplest casesofthreeandfour-dimensionaldynamicalsystems,thatwereob- tained throughthelinearchaintrickfromtheKaldor-Kaleckigrowthmodel with distributeddelay.Forbothmodels,theexistenceconditions'oftheHopf bifurcation withrespecttothetimedelayparameterandtherateofgrowthpa- rameter areestablished.Finally,wediscussthenumericalstudyintheprevious papers