Solitary states for coupled oscillators with inertia.

Networks of identical oscillators with inertia can display remarkable spatiotemporal patterns in which one or a few oscillators split off from the main synchronized cluster and oscillate with different averaged frequency. Such "solitary states" are impossible for the classical Kuramoto model with sinusoidal coupling. However, if inertia is introduced, these states represent a solid part of the system dynamics, where each solitary state is characterized by the number of isolated oscillators and their disposition in space. We present system parameter regions for the existence of solitary states in the case of local, non-local, and global network couplings and show that they preserve in both thermodynamic and conservative limits. We give evidence that solitary states arise in a homoclinic bifurcation of a saddle-type synchronized state and die eventually in a crisis bifurcation after essential variation of the parameters.

Chimera complexity.

We show an amazing complexity of the chimeras in small networks of coupled phase oscillators with inertia. The network behavior is characterized by heteroclinic switching between multiple saddle chimera states and riddling basins of attractions, causing an extreme sensitivity to initial conditions and parameters. Additional uncertainty is induced by the presumable coexistence of stable phase-locked states or other stable chimeras as the switching trajectories can eventually tend to them. The system dynamics becomes hardly predictable, while its complexity represents a challenge in the network sciences.

Smallest chimera states.

We demonstrate that chimera behavior can be observed in small networks consisting of three identical oscillators, with mutual all-to-all coupling. Three different types of chimeras, characterized by the coexistence of two coherent oscillators and one incoherent oscillator (i.e., rotating with another frequency) have been identified, where the oscillators show periodic (two types) and chaotic (one type) behaviors. Typical bifurcations at the transitions from full synchronization to chimera states and between different types of chimeras have been described. Parameter regions for the chimera states are obtained in the form of Arnold tongues, issued from a singular parameter point. Our analysis suggests that chimera states can be observed in small networks relevant to various real-world systems.