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Chimera states in networks of Van der Pol oscillators with hierarchical connectivities

Ulonska, Stefan; Omelchenko, Iryna; Zakharova, Anna; Schöll, Eckehard

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We analyse chimera states in ring networks of Van der Pol oscillators with hierarchical coupling topology. We investigate the stepwise transition from a nonlocal to a hierarchical topology and propose the network clustering coefficient as a measure to establish a link between the existence of chimera states and the compactness of the initial base pattern of a hierarchical topology; we show that a large clustering coefficient promotes the occurrence of chimeras. Depending on the level of hierarchy and base pattern, we obtain chimera states with different numbers of incoherent domains. We investigate the chimera regimes as a function of coupling strength and nonlinearity parameter of the individual oscillators. The analysis of a network with larger base pattern resulting in larger clustering coefficient reveals two different types of chimera states and highlights the increasing role of amplitude dynamics. Chimera states are an example of intriguing partial synchronization patterns appearing in networks of identical oscillators. They exhibit a hybrid structure combining coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics.1,2 Recent studies have demonstrated the emergence of chimera states in a variety of topologies and for different types of individual dynamics. In this paper, we analyze chimera states in networks with complex coupling topologies arising in neuroscience. We provide a systematic analysis of the transition from nonlocal to hierarchical (quasi-fractal) connectivities in ring networks of identical Van der Pol oscillators and use the clustering coefficient and the symmetry properties to classify different topologies with respect to the occurrence of chimera states. We show that symmetric connectivities with large clustering coefficients promote the emergence of chimera states, while they are suppressed by slight topological asymmetries or small clustering coefficient.
Published in: Chaos: An Interdisciplinary Journal of Nonlinear Science, 10.1063/1.4962913, American Institute of Physics (AIP)
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