@phdthesis{Omelchenko2021,
  author    = {Omelchenko, Oleh},
  title     = {Synchronit{\"a}t-und-Unordnung-Muster in Netzwerken gekoppelter Oszillatoren},
  doi       = {10.25932/publishup-53596},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-535961},
  school      = {Universit{\"a}t Potsdam},
  pages     = {152},
  year      = {2021},
  abstract  = {Synchronization of coupled oscillators manifests itself in many natural and man-made systems, including cyrcadian clocks, central pattern generators, laser arrays, power grids, chemical and electrochemical oscillators, only to name a few. The mathematical description of this phenomenon is often based on the paradigmatic Kuramoto model, which represents each oscillator by one scalar variable, its phase. When coupled, phase oscillators constitute a high-dimensional dynamical system, which exhibits complex behaviour, ranging from synchronized uniform oscillation to quasiperiodicity and chaos. The corresponding collective rhythms can be useful or harmful to the normal operation of various systems, therefore they have been the subject of much research. Initially, synchronization phenomena have been studied in systems with all-to-all (global) and nearest-neighbour (local) coupling, or on random networks. However, in recent decades there has been a lot of interest in more complicated coupling structures, which take into account the spatially distributed nature of real-world oscillator systems and the distance-dependent nature of the interaction between their components. Examples of such systems are abound in biology and neuroscience. They include spatially distributed cell populations, cilia carpets and neural networks relevant to working memory. In many cases, these systems support a rich variety of patterns of synchrony and disorder with remarkable properties that have not been observed in other continuous media. Such patterns are usually referred to as the coherence-incoherence patterns, but in symmetrically coupled oscillator systems they are also known by the name chimera states. The main goal of this work is to give an overview of different types of collective behaviour in large networks of spatially distributed phase oscillators and to develop mathematical methods for their analysis. We focus on the Kuramoto models for one-, two- and three-dimensional oscillator arrays with nonlocal coupling, where the coupling extends over a range wider than nearest neighbour coupling and depends on separation. We use the fact that, for a special (but still quite general) phase interaction function, the long-term coarse-grained dynamics of the above systems can be described by a certain integro-differential equation that follows from the mathematical approach called the Ott-Antonsen theory. We show that this equation adequately represents all relevant patterns of synchrony and disorder, including stationary, periodically breathing and moving coherence-incoherence patterns. Moreover, we show that this equation can be used to completely solve the existence and stability problem for each of these patterns and to reliably predict their main properties in many application relevant situations.},
  language  = {en}
}