Smallest chimeras under repulsive interactions

Abstract: We present an exemplary system of three identical oscillators in a ring interacting repulsively to show up chimera patterns. The dynamics of individual oscillators is governed by the superconducting Josephson junction. Surprisingly, the repulsive interactions establish a symmetry of compelete synchrony in the ring, which is broken with increasing interactions when the junctions pass through serials of asynchronous states (periodic and chaotic), but finally emerge into chimera states. The chimera pattern appears in chaotic rotational motion of the three junctions when two junctions evolve coherently while the third junction is incoherent. For larger repulsive coupling, the junctions evolves into another chimera pattern in a periodic state when two junctions remain coherent in rotational motion and one transits to incoherent librational motion. This chimera pattern is sensitive to initial conditions, in the sense, that the chimera state flips to another pattern when two junctions switch to coherent librational motion and the third junction remains in rotational motion, but incoherent. The chimera patterns are detected by using partial and global error functions of the junctions while the librational and rotational motions are identified by a libration index. All the collective states, complete synchrony, desynchronization and two chimera patterns, are delineated in a parameter plane of the ring of junctions, where the boundaries of complete synchrony are demarcated by using the master stability function.