A remark on geometric desingularization of a non-hyperbolic point using hyperbolic space

摘要

Asteadystate(orequilibriumpoint)ofadynamicalsystemishyperboliciftheJacobianatthesteadystatehasnoeigenvalueswithzerorealparts.Inthiscase,thelinearizedsystemdoesqualitativelycapturethedynamicsinasmallneighborhoodofthehyperbolicsteadystate.However,oneisoftenforcedtoconsidernon-hyperbolicsteadystates,forexampleinthecontextofbifurcationtheory.Ageometrictechniquetodesingularizenon-hyperbolicpointsistheblow-upmethod.Theclassicalcaseofthemethodismotivatedbydesingularizationtechniquesarisinginalgebraicgeometry.Theideaistoblowupthesteadystatetoasphereoracylinder.Intheblown-upspace,oneisthenoftenabletogainadditionalhyperbolicityatsteadystates.Themethodhasalsoturnedouttobeakeytooltodesingularizemultipletimescaledynamicalsystemswithsingularities.Inthispaper,wediscussanexplicitexampleoftheblow-upmethodwherewereplacethesphereintheblow-upbyhyperbolicspace.Itisshownthatthecalculationsworkinthehyperbolicspacecaseasforthesphericalcase.Thisapproachmaybeevenslightlymoreconvenientifonewantstoworkwithdirectionalcharts.Hence,itisdemonstratedthatthesphereshouldbeviewedasanauxiliaryobjectintheblow-upconstruction.Othersmoothmanifoldsarealsonaturalcandidatestobeinsertedatsteadystates.Furthermore,weconjectureseveralproblemswherereplacingthespherecouldbeparticularlyuseful,i.e.,inthecontextofsingularitiesofgeometricflows,foravoidingcompactification,andgenerating'interior'steadystates...