Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems

Abstract

This chapter is devoted to the study of generic bifurcations of fixed points of two-parameter maps. First we derive a list of such bifurcations. As for the final two bifurcations in the previous chapter, the description of the majority of these bifurcations is incomplete in principle. For all but two cases, only approximate normal forms can be constructed. Some of these normal forms will be presented in terms of associated planar continuous-time systems whose evolution operator $\phi ^1$ approximates the map in question (or an appropriate iterate of the map). We present bifurcation diagrams of the approximate normal forms and discuss their relationships with the original maps.