Smallest chimera states

Abstract

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...

Nonlinearity of local dynamics promotes multi-chimeras

“…However, it has been shown for phase oscillator networks that time delay can also induce multi-chimeras 23 . The existence of chimera states in sys-tems with time-delayed couplings has been also reported in 50,[63][64][65][66] . In particular, for coupled phase oscillator systems it has been found that chimeras are robust to small time delays and delay distributions 64 and can become unstable depending on the value of delay 65 .…”

Symmetry broken states in an ensemble of globally coupled pendulums

“…Different kinds of chaotic attractors with partially broken symmetry have been investigated there beyond phase oscillator systems. The authors of the article [25] have described regular and chaotic chimera states in the system of three coupled phase oscillators with inertia. These states have been detected in experiments with mechanical systems [26,27].…”

Blinking chimeras in globally coupled rotators

“…Furthermore, examples of chimera regimes within rather small sets of oscillators have been reported 3 . Recent studies have revealed that chimera states can be quite complex.…”