Networks of phase oscillators: synchronization and applications - Celso Bernardo da Nóbrega de Freitas (2017)
This work explores synchronization regarding networks of active units. More specifically, we focus on the Kuramoto Model (KM), which is one of the most successful models for collective behavior. Agents here are modeled as phase-oscillators, in the sense that they are represented by a unidimensional state with a 2$π$ increment for every complete cycle. Such model is remarkably important due to its relative simplicity and wide range of applications, either as one of its variations or as a building block for other systems. Given a time series obtained from an oscillatory phenomenon, phase assignment is the name of the process of choosing phase-variables for it. The first contribution (I) of this thesis is a test bed to evaluate phase assignment methodologies: the Double Strip Test Bed (DSTB). This is done by defining a chaotic oscillator surrogate by embedding phase-variable from a KM into suitable three dimensional surface. DSTB allows comparison between methods of phase assignment for time series since it provides an a priori reference phase-variables. For the second contribution (II), we introduce a generalization of the KM: the Deserter Hubs Model (DHM). It corresponds to a non-linear coupling scheme, where oscillators can shift from conformist to contrarian under the influence of a sufficiently large number of neighbors. This scheme holds analogy with neural synchronous oscillations at Parkinson disease. Therefore, we were able to: (i) give sufficiently conditions for phase locking; (ii) numerically show several qualitative behaviors; and (iii) correlate some of them with metrics from the corresponding coupling graph. The last contribution (III) deals with the classic version of KM, introducing a new question: Does the position of non-identical oscillators into the nodes of a graph affect synchronization? In particular, we are interested in homophily/heterophily configurations, which corresponds to multi-agents systems whose units tend to bond with others with similar/dissimilar characteristic in comparison with themselves. Thus, we present numerical evidences that Similar patterns favor the emergence of synchronization for small coupling parameter, while Dissimilar patterns undergoes abrupt synchronization for larger coupling values.