Types of change of stability and corresponding types of bifurcations

The general topic is the connection between a change of stability of an equilibrium point or invariant set $M$ of a (semi-) dynamical system depending on a parameter and a bifurcation of $M$ (generalizing the Hopf bifurcation). In particular, we address the case where $M$ is unstable (for instance a saddle) for a certain value $\lambda_0$ of a parameter $\lambda$, and stable for certain nearby values. Two kinds of bifurcations are considered: "extracritical", i.e. splitting of the set $M$ as $\lambda$ passes the value $\lambda_0$, and "critical" (also called "vertical"), a term which refers to the accumulation of closed invariant set at $M$ for $\lambda=\lambda_0$. Also, two kinds of change of stability are considered, corresponding to the presence or absence of a certain generalized equistability property for $\lambda\ne\lambda_0$. Connections are established between the type of change of stability and the types of bifurcation arising from them.