Summary
We review partial synchronization patterns emerging in networks of nonlinear oscillators. An intriguing example are chimera states which consist of spatially coexisting domains of coherent (synchronized) and incoherent (desynchronized) dynamics, i.e., seemingly incongruous parts. We show that a plethora of partial synchronization patterns arise if one goes beyond the simple Kuramoto phase oscillator model, and considers coupled phase and amplitude dynamics, and more complex topologies than one-dimensional regular ring network, e.g., small world or fractal connectivities or multilayer structures. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator various space-time patterns including amplitude mediated phase chimeras, amplitude chimeras, chimera death, double chimeras, and solitary states occur. We also focus on the role of time-delayed coupling in controlling these patterns, and on the subtle interplay of local dynamics, delay, and the network structure. Chimera states might be of relevance in inducing and terminating epileptic seizures, or in unihemispheric sleep which is found in certain migratory birds and mammals, or in cognitive functions (learning, memory) of the brain where certain areas act as relay.
