October  1999, 5(4): 741-752. doi: 10.3934/dcds.1999.5.741

Types of change of stability and corresponding types of bifurcations

1. 

Universidad Autónoma Metropolitana, Unidad Iztapalapa, Av. Michoacán y la Purísima, Apdo. Postal 55-534, Mexico 09340, D.F., Mexico, Mexico

Received  August 1998 Revised  May 1999 Published  July 1999

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.
Citation: L. Aguirre, P. Seibert. Types of change of stability and corresponding types of bifurcations. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 741-752. doi: 10.3934/dcds.1999.5.741
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