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In this work, we consider a ring of coupled electronic (Wien-bridge) oscillators from a perspective combining modeling, simulation, and experimental observation. Following up on earlier work characterizing the pairwise interaction of Wien-bridge oscillators by Kuramoto-Sakaguchi phase dynamics, we develop a lattice model for a chain thereof, featuring an exponentially decaying spatial kernel. We find that for certain values of the Sakaguchi parameter a, states of traveling phasedomain fronts<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1063/1.5009088">doi:10.1063/1.5009088</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/29092454">pmid:29092454</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/skjjdfrgnng45ditt5fqwq2pwm">fatcat:skjjdfrgnng45ditt5fqwq2pwm</a> </span>
more »... olving the coexistence of two clearly separated regions of distinct dynamical behavior, can establish themselves in the ring lattice. Experiments and simulations show that stationary coexistence domains of synchronization only manifest themselves with the introduction of a local impurity; here an incoherent cluster of oscillators can arise reminiscent of the chimera states in a range of systems with homogeneous oscillators and suitable nonlocal interactions between them. Published by AIP Publishing. https://doi.org/10.1063/1.5009088 Mass-synchronization in biological systems has been studied extensively for several decades now using mathematical tools. The basic question is: how do individual, non-identical oscillators (neurons, fireflies, etc.) influence each other's dynamics to give rise to group synchronization on the global scale. Yoshiki Kuramoto's discovery that the synchronized state can arise via a second-order phase transition from the incoherent state led to a surge of interest in the field, as it opened up many new lines of investigation. What role, for instance, did the coupling topology play in the emergence of this temporal phase transition? More recently it was predicted that an intriguing hybrid state could stabilize itself in a lattice of identical oscillators, where a synchronized cluster could coexist with an incoherent cluster, and it soon began to be called the chimera state. While this state was initially solely a numerical and then an analytical prediction, more recently some experiments have verified its existence in a variety of systems. In many experimental contexts it is somewhat difficult to satisfy or even approximate the constraints imposed by the original theory, as it relates to the spatial coupling kernel, for instance, and some of the observations have also been indirect. Now there have been some theoretical reports of traveling domain fronts in non-locally coupled Kuramoto-type lattices. Here we report on direct observations of such traveling fronts, as well as impurityinduced chimera-like states in a lattice of 32 electrical self-oscillators. We also conduct a numerical exploration of the system's approximate dynamics in good agreement with experimental observations. While our system does not appear to lend itself to true chimera states in the case of identical oscillators, it provides a promising example of nonlocal interactions in an electrical lattice that could be interesting to further explore (including in higher dimensional settings) as a rich dynamical example of pattern formation in a setting amenable to detailed spatiotemporal probing.
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