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.

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 realworld systems.Chimera states are spatiotemporal patterns consisting of spatially separated domains of coherent (synchronized) and incoherent (desynchronized) behavior, which appear in the networks of identical units. The original discovery in a network of phase oscillators [1][2][3] has sparked a tremendous activity of first theoretical studies [4][5][6][7][8][9][10][11][12] and next experimental observations [13][14][15][16][17][18]. In real-world systems, chimera states might play role in understanding of complex behavior in biological (modular neural networks [19], the unihemispheric sleep of birds and dolphins [20], epileptic seizures [21]), engineering (power grids [22,23]) and social [24] systems. More references can be found in two recent review papers [25,26].Chimera states are typically observed in the large networks of different topologies, but recently it has been suggested that they can also be observed in small networks [27][28][29][30][31]. Ashwin & Burylko [27] have defined a weak chimera state as one referring to a trajectory in which two or more oscillators are frequency synchronized and one or more oscillators drift in phase and oscillate with different mean frequency with respect to the synchronized group. First, it has been observed that these states can exist in small networks of as few as 4 phase oscillators [27][28][29][30] and also in the model of semiconductor lasers [31]. Experimentally, chimera states of this type have been recently observed in small networks of optoelectronic oscillators [32] as well as coupled pendula [33,34].In the Letter, we show that the weak chimera patterns which are characterized by two frequency synchronized oscillators and one evolving with different frequency can be observed in the networks of 3 identical nodes. As the proof of the concept we use a network of Kuramoto oscillators with inertia. We identify three different types of chimeras, namely (i) in-phase chimeras in which coherent oscillators are phase synchronized and incoherent one rotates with a different frequency, (ii) anti-phase chimeras in which coherent oscillators alternates with respect to each other and incoherent one oscillates with a different frequency, (iii) chaotic chimeras in which two oscillators are synchronized...

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of co-existing coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: 1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, 2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and 3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

The stability of synchronised networked systems is a multi-faceted challenge for many natural and technological fields, from cardiac and neuronal tissue pacemakers to power grids. In the latter case, the ongoing transition to distributed renewable energy sources is leading to a proliferation of dynamical actors. The desynchronization of a few or even one of those would likely result in a substantial blackout. Thus the dynamical stability of the synchronous state has become a focus of power grid research in recent years.In this letter we uncover that the non-linear stability against large perturbations is dominated and threatened by the presence of solitary states in which individual actors desynchronise. Remarkably, when taking physical losses in the network into account, the back-reaction of the network induces new exotic solitary states in the individual actors, and the stability characteristics of the synchronous state are dramatically altered. These novel effects will have to be explicitly taken into account in the design of future power grids, and their existence poses a challenge for control.While this letter focuses on power grids, the form of the coupling we explore here is generic, and the presence of new states is very robust. We thus strongly expect the results presented here to transfer to other systems of coupled heterogeneous Newtonian oscillators.

We demonstrate for a nonlinear photonic system that two highly asymmetric feedback delays can induce a variety of emergent patterns which are highly robust during the system's global evolution. Explicitly, two-dimensional chimeras and dissipative solitons become visible upon a space-time transformation. Switching between chimeras and dissipative solitons requires only adjusting two system parameters, demonstrating self-organization exclusively based on the system's dynamical properties. Experiments were performed using a tunable semiconductor laser's transmission through a Fabry-Perot resonator resulting in an Airy function as nonlinearity. Resulting dynamics were band-pass filtered and propagated along two feedback paths whose time delays differ by two orders of magnitude. An excellent agreement between experimental results and theoretical model given by modified Ikeda equations was achieved.