Bifurcationswithconstraintsareopenproblemsappearedinresearchonperiodicbifurcationsofnonlineardynamicalsystems,butthepresentsingularitytheorydoesn'tcontainanyanalyticalmethodsandresultsaboutit.Asthecomplementtosingularitytheoryandthefirststeptostudyonconstrainedbifurcations,herearegiventhetransitionsetsandpersistentperturbedbifurcationdiagramsof10elementarybifurcationofcodimensionnomorethanthree.