Codimension 2 Bifurcations of Iterated Maps

Abstract

This thesis investigates some properties of discrete-time dynamical systems, generated by iterated maps. In particular we study local bifurcations where two parameters are essential to describe the dynamical properties of the system near a fixed point or a cycle. There are 11 such cases. Knowledge of these bifurcations is important as they form boundary corners of stability regions, which is an important issue when one investigates properties of dynamical systems. In chapter two, we first review some aspects of codimension 2 bifurcations which have been studied before. After this short summary of the involved bifurcation curves and scenarios, three cases are analyzed, which are understood less, namely the fold-flip, flip-Neimark-Sacker and double Neimark-Sacker bifurcations. Parameter-dependent normal forms in the minimal possible phase dimension are given up to a certain sufficient degree. Higher order terms are neglected at first. Then local and global bifurcations of these truncated normal forms are investigated. Finally, the effect of truncation of higher order terms in the normal form is discussed. The bifurcation analysis is representative, but the dynamics of the normal form is in general not exactly the same as in the original system. The effect of certain perturbations is investigated numerically. For bifurcations of invariant curves Chenciner, Takens and Wagener, and Broer et.al. have obtained results, which we confirm, connect and extend. In chapter three we discuss codimension two bifurcations of maps in an arbitrary, but finite number of dimensions. Formulas are derived for the coefficients of the normal form with the aid of Center Manifold Reduction. These coefficients can be used to draw conclusions about the dynamics of the original systems. An implementation of this approach in the continuation package MATCONT is also described. It is capable of continuation of cycles and codimension one bifurcation curves of cycles in one and two parameters, respectively. During the continuation bifurcations can be detected and coefficients for the normal forms are computed automatically. Another new feature compared to existing software, is the possibility to switch to branches to certain bifurcation curves emanating from codimension two points. In chapter four, several examples from control, population dynamics and meteorology are discussed to illustrate the new results and the developed methods.