Abstract
Coupled oscillators, even identical ones, display a wide range of behaviours, among them synchrony and incoherence. The 2002 discovery of so-called chimera states, states of coexisting synchronized and unsynchronized oscillators, provided a possible link between the two and definitely showed that different parts of the same ensemble can sustain qualitatively different forms of motion. Here, we demonstrate that globally coupled identical oscillators can express a range of coexistence patterns more comprehensive than chimeras. A hierarchy of such states evolves from the fully synchronized solution in a series of cluster-splittings. At the far end of this hierarchy, the states further collide with their own mirror-images in phase space – rendering the motion chaotic, destroying some of the clusters and thereby producing even more intricate coexistence patterns. A sequence of such attractor collisions can ultimately lead to full incoherence of only single asynchronous oscillators. Chimera states, with one large synchronized cluster and else only single oscillators, are found to be just one step in this transition from low- to high-dimensional dynamics.
Highlights
Coupled oscillators, even identical ones, display a wide range of behaviours, among them synchrony and incoherence
Ensembles of coupled oscillators are one class of apparently simple dynamical systems that yet may adopt states ranging from full synchrony to complete incoherence, and which has provided insights in virtually any discipline, ranging from the natural sciences to sociology[6,7]
Chimera states are just one of a multitude of coexistence patterns, all consisting of clusters, that is, internally synchronized groups of oscillators, of widely different sizes and dynamics, and possibly including one or several single oscillators
Summary
Even identical ones, display a wide range of behaviours, among them synchrony and incoherence. Chimera states are just one of a multitude of coexistence patterns, all consisting of clusters, that is, internally synchronized groups of oscillators, of widely different sizes and dynamics, and possibly including one or several single oscillators. Each variant collides with some of its mirror-images, creating larger attractors with higher symmetry This blows up some of the clusters, the resulting single oscillators moving on average. The nonlinear global coupling in Eq (1) stands out by featuring two qualitatively different chimera states, each of them deduced to somehow emerge from a corresponding type of two-cluster solution[24] This coupling was inspired by electrochemical experiments, wherein the oxide layer on a silicon electrode displays a wide range of spatiotemporal patterns[17]. A few experimental measurements reminiscent of new results in Eq (1)
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