Symmetric States Requiring System Asymmetry

Abstract

Spontaneous synchronization has long served as a paradigm for behavioral uniformity that can emerge from interactions in complex systems. When the interacting entities are identical and their coupling patterns are also identical, the complete synchronization of the entire network is the state inheriting the system symmetry. As in other systems subject to symmetry breaking, such symmetric states are not always stable. Here we report on the discovery of the converse of symmetry breaking-the scenario in which complete synchronization is not stable for identically-coupled identical oscillators but becomes stable when, and only when, the oscillator parameters are judiciously tuned to nonidentical values, thereby breaking the system symmetry to preserve the state symmetry. Aside from demonstrating that diversity can facilitate and even be required for uniformity and consensus, this suggests a mechanism for convergent forms of pattern formation in which initially asymmetric patterns evolve into symmetric ones.Symmetry-the property of appearing the same from different viewpoints-is so central to physics that Weyl [1] suggested that "all a priori statements in physics have their origin in symmetry"; Anderson [2] went further to propose that "physics is the study of symmetry." In the study of complex networks this tradition was for many years relegated to a secondary position, for the excellent reason that real complex systems appeared not to exhibit symmetries. Recent work has shown, however, that they not only can exhibit a myriad of symmetries [3] but also that such symmetries have direct implications for dynamical behavior (see Ref.[4] for example). Partially motivated by that, significant recent attention has been dedicated to the extreme, most symmetric case of uniform networks in which nodes are all identically coupled to the others and have no natural grouping, as in a ring or all-to-all network. It has been shown that such systems can exhibit spatiotemporal patterns of coexisting synchronous and non-synchronous behavior [5,6], for which elaborated mathematical analysis techniques are now available [7]. The emergence of these patterns can be regarded as a form of symmetry breaking, since the realized state has less symmetry than the system [8].Here we demonstrate for the first time that the converse of symmetry breaking with the roles of the system and its state reversed-which we term asymmetryinduced symmetry-is also possible. We provide examples of uniform, rotationally symmetric networks of coupled oscillators for which stable uniform states (thus rotationally symmetric states) do not exist when the nodes are identical but do exist when the nodes are not identical.In a network of coupled oscillators a uniform, symmetric state represents synchronization, in which all units swing in concert, following the exact same dynamics as a function of time [9]. Synchronization dynamics is widespread across fields-ranging from physics and engineering to biology and social sciences-and is intimately related to the twin proce...

Topological Control of Synchronization Patterns: Trading Symmetry for Stability

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Critical Switching in Globally Attractive Chimeras

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Stabilization of mode in imbalanced operation of matrix converter by time‐delayed feedback control

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