Symmetry-Breaking bifurcations and reservoir computing in regular oscillator networks

André Röhm, Technische Universität Berlin, Technische Universität Berlin, Kathy Lüdge
2019
The focus of this thesis is the investigation of regular oscillator networks with numerical and analytical tools. Such systems are ubiquitous in nature and can emerge in the shape of vastly different processes. Among others, they can be mathematically derived as discrete approximations of continuous wave equations or as a local approximation of coupled dynamical systems. In the most basic meaning of the word, an oscillator is a system that shows a periodic change in one of its state variables.
more » ... his thesis will focus on two different mathematical models: The Stuart-Landau oscillator that is popular in bifurcation theory and the more applied Lang-Kobayashi model for laser dynamics. Lasers can be seen as optical oscillators and they are most accessible of all oscillatory systems for precise and fast experiments. Many previous works have covered different aspects of coupled oscillators, with some effects such as synchronization having been studied in some form for more than three centuries. The overall literature on the topic of coupling-induced dynamics is vast, and many different subtopics can be identified, such as the influence of delay, the application of control schemes, the properties of biological and artificial neural networks, the causes of chaotic motion or the importance of stochastic processes. The focus in this thesis will be on highly symmetric networks of coupled oscillators, where the local dynamics follow very simple models. The focus is therefore on the coupling-induced effects in general and not on elaborate and extremely detailed descriptions of specific experiments. However, because the effects of coupling and especially those connected to symmetries are in some sense universal, generic models for oscillators and lasers more than suffice for the investigations done here. A particular focus will be on the emergence of symmetry-broken states in rings of oscillators. Several different novel aspects of these symmetry-breaking mstates are discussed and their connections to the established solutions f [...]
doi:10.14279/depositonce-8246 fatcat:3cgtqhfmibaipcttr5prevd2b4