Bifurcations of Maps in the Software Package CONTENT

Abstract

The qualitative behaviour of iterates of a map can be very complicated. One approach to these phenomena starts with the simplest situation, the case where the map has a fixed point. Under parameter variations, the fixed point typically moves until a bifurcation value is reached and one of three possible more complex phenomena is encountered. These are fold, flip and Neimark - Sacker bifurcations; they are called codimension one phenomena because they generically appear in problems with one free parameter.

The software package CONTENT (continuation environment) combines numerical methods (integration, numerical continuation etcetera) with symbolic methods (e.g. symbolic derivatives) and allows (among other things) to numerically continue fixed points and to detect, compute and continue fold points, flip points and Neimark - Sacker points. To the best of our knowledge content is the only softwaxe that allows to detect and compute all codimension two points on such curves, including strong resonances and degenerate Neimark - Sacker bifurcations. The paper provides details on defining systems and test functions implemented in content for these purposes.

We show the power of the software by studying the behaviour of an electromechanical device that exhibits a complicated bifurcation behaviour, the so - called Sommerfeld’s efFect. In this example the map is defined by the time integration of a three - dimensional dynamical system over a fixed time interval.