Symmetry, Synchronization, and Switching Chimeras

Zhang, Yuanzhao

doi:https://doi.org/10.21985/n2-5p0r-de83

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Symmetry, Synchronization, and Switching Chimeras

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There is a rich history on the study of the interplay between symmetry and synchronization in networks. At the most fundamental level, many synchronization patterns are induced by underlying network symmetries. However, when stability is taken into account, the relation between symmetry and synchronization is far from monotonic. In this dissertation, we first demonstrate that introducing asymmetry in a network can often improve its synchronizability. Such asymmetry can reside in either the network structure or the node proprieties. In both cases, we characterize the unexpected positive effect of asymmetry on synchronization through theory, simulations, and experiments. On the other hand, even networks with the highest degree of symmetry can support very complex synchronization patterns. One prominent example is the so-called chimera states---patterns of synchrony representing coexistence of coherence and incoherence among identically coupled identical oscillators. Here, we give a detailed characterization of switching chimeras, which are chimera states that are robust and sensitive to noise at the same time. We also establish incoherence-stabilized coherence as a model-free mechanism that gives rise to chimeras in a wide range of systems. The phenomena above provide valuable insights into the intricate relation between symmetry and synchronization, and they call for a general framework to analyze network synchronization that is not encumbered by symmetry considerations. Utilizing the theory of matrix-$*$ algebra, we develop such a symmetry-independent framework, which leads to fast and versatile algorithms that can characterize the stability of arbitrary synchronization patterns on networks. This framework is then further extended to treat dynamical processes in complex systems with generalized interactions, including hypergraphs, simplicial complexes, multilayer networks, and temporal networks.

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