Structural properties of the nested period incrementing bifurcation scenario

Abstract

The importance of piecewise-smooth and especially that of discontinuous system models is well-established. One of the properties of these systems is the possibility of border collision bifurcations, which can form complex bifurcation scenarios. In this thesis, the description of the recently discovered nested period incrementing bifurcation scenario is significantly extended to form a more complete understanding of its topological structure. It is shown that the scenario is governed by codimension-two big bang bifurcations, in which well organised families of periodic orbits appear. The symbolic description of these families is determined by the unstable periodic orbits of the investigated system. This work introduces concise rules based on symbolic dynamics, by which the structure of the bifurcation scenario in the two-dimensional parameter space is fully described. It is furthermore shown that the results are easily transferred to discontinuous piecewise-linear systems defined on $n$ partitions, in general.