On Bifurcations Arising from Unstable Equilibria and Invariant Sets

It is proved that whenever an isolated compact invariant set (or equilibrium point)Mis unstable for a certain valueλ0of a parameterλand stable for values ofλclose toλ0,Mundergoes a bifurcation atλ0. The setting is a locally asymptotically compact family of semidynamical systems on a metric space