Some Aspects on Global Analysis of Discrete Time Dynamical Systems

Dynamical systems theory distinguishes two types of bifurcations: those which can be studied in a small neighborhood of an invariant set (local) and those which cannot (global). In contrast to local bifurcations, global ones cannot be investigated by a Taylor expansion, neither they are detected by purely performing stability analysis of periodic points. Global bifurcations often occur when larger invariant sets of the system collide with each other or with other fixed points/cycles. This chapter focuses on several aspects of global bifurcation analysis of discrete time dynamical systems, covering homoclinic bifurcations as well as inner and boundary crises of attracting sets.