Abstract
Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are not only of intrinsic interest but also of very wide importance. In 1975, Kuramoto proposed an analytically tractable model to describe these systems, which has since been successfully applied in many contexts and remains a subject of intensive research. Some related problems, however, remain unclarified for decades, such as the existence and properties of the oscillator glass state. Here we present a detailed analysis of a very general form of the Kuramoto model. In particular, we find the conditions when it can exhibit glassy behaviour, which represents a kind of synchronous disorder in the present case. Furthermore, we discover a new and intriguing phenomenon that we refer to as super-relaxation where the oscillators feel no interaction at all while relaxing to incoherence. Our findings offer the possibility of creating glassy states and observing super-relaxation in real systems.
Highlights
Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are of intrinsic interest and of very wide importance
The practical usefulness of these studies stems from widespread abundance of user-defined systems that can be described by the Kuramoto model (KM), for example, the above-mentioned laser arrays[4,5] and Josephson junction circuits[8], where one can adjust the coupling between the oscillators quite freely
We discover new states in the coupled oscillator populations, which are in some sense similar to physical glasses, and we establish the conditions for their appearance (equations (36) and (37))
Summary
Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are of intrinsic interest and of very wide importance. The Kuramoto model (KM)[1] was introduced and developed to provide an analytically tractable description of the populations of coupled phase oscillators that so often appear in real life It has been applied successfully in many fields[2,3], for example, to describe the collective behaviour of lasers[4,5], neurons in the brain[6,7], Josephson junction arrays[8] and even humans[9]. Many systems, including networks of interacting spins[26,27], dipoles[28], and electrons[29], have been found to display behaviour reminiscent of a glass structure[30], and it has been suggested[25] that populations of coupled oscillators can demonstrate glassy states of some kind. It implies that there may be cases where it is, in principle, impossible to infer the underlying coupling structure from the observed data
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