Simulation and analysis of complex dynamics in a low-dimensional model of a vibration-machine

Abstract

In the past decades, the importance of discontinuous piecewise-smooth models has been well-established. So far, the dynamics of such models are mostly analyzed based on functions with a single point of discontinuity. This class of functions is well-investigated in the literature and many theoretical results are known. Even though many physical phenomena can be described by this class, for more complex systems, a single point of discontinuity is not sufficient. In this thesis, this theory is extended by the analysis of a map in which the number of discontinuities varies based on different parameters. It is shown that this map does for integer-valued parameters result in clear dynamics, e.g. a sausage-shaped period-adding cascade. Even though these structures are already known in the literature, the understanding of its dynamics is key for the analysis of the structures for a more generalized range of parameter values. For example, for real-valued parameters the map show a much more complex structure consisting of a combination of period-adding and period-incrementing with different coexistence scenarios. Furthermore, it is shown that the iteratively increasing of a parameter related to the number of discontinuities destroys the sausage-shaped structure and transforms it for the next higher integer value to a more complex sausage-shaped structure. It is investigated how this structure is destroyed and how it evolves in the intermediate steps. Additionally, in this new structure an example configuration of the model is found where well-established results from a map with one discontinuity can not be applied.