Final Report Summary - GECO (Dynamics of Phase Oscillator Networks with Generalized Coupling) The interaction of units which undergo repetitive motion is one of the fundamental schemes in nature and technology. For example these oscillations are crucial to make our hearts beat: all heart cells have to contract together at the same time. Typically, the interaction between units is assumed to be simple whereas more complicated interaction has been neglected. By contrast, over its duration, GECO has yielded many deep insights into the dynamics of phase oscillator networks with generalized rather than simple coupling.In the first year, progress was made to understand the dynamics of small oscillator networks with generalized coupling. On the one hand, insights into the dynamics and bifurcations of small, fully symmetric populations of phase oscillators were obtained. On the other hand, a significant step was made to understand structured networks with generalized coupling: we proved a general result on the appearance of localized frequency synchrony with chaotic dynamics (chaotic weak chimeras). This closes the gap between nonchaotic weak chimeras and ‘classical’ chimeras which show chaotic dynamics in finite networks.The focus of GECO diversified in the second year. For small networks, we showed how nonpairwise generalized coupling (consisting of nonlinear functions that depend not just on pairs but of three or more phase variables) breaks the degeneracy of Kuramoto-type systems to give rise to chaotic dynamics. For the continuum of infinitely many oscillators, we uncovered that networks with coupling through generalized phase lags (for example through transmission delays) give rise not only to locally synchronized dynamics but also to chaotic mean field fluctuations. Finally, in work that went well beyond the originally planned we showed that combining higher harmonics and nonpairwise coupling yields global dynamical phenomena.During the course of GECO, the fellow established collaborations with experimentalists to connect the theoretical progress in idealized phase oscillator networks with real-world oscillator networks. We demonstrated localized frequency synchrony in an experimental setup of electrochemical oscillators. These results give crucial insights how generalized coupling shapes the dynamics of real-world oscillator networks which may be used in a wide range of applications in the future.Further information and relevant publications can be found on the project’s website http://ex.ac.uk/geco .