3 years ago

Symmetry and symmetry breaking in coupled oscillator communities.

Per Sebastian Skardal

We study the role of symmetry and asymmetry in the collective dynamics in systems of two communities of coupled phase oscillators with different natural frequency distributions. In the case where the communities' dynamics are symmetric we derive the system's bifurcation curves using analytical tools, classifying the macroscopic system dynamics as a function of the intra- and inter-community coupling strengths as well as the the difference in the means of the communities' frequency distributions. This analysis describes the bifurcations between incoherence, standing wave, and partially synchronized states and reveals bistability regions between these states. We find that the heterogeneity in the coupling strengths, i.e., the difference in intra- and inter-community coupling, allows for bistability to occur for any choice of frequency distributions, provided that the means of the two distributions are unequal. Moreover, as the difference between frequency distributions increases, bistability is promoted, resulting in larger regions of parameter space that display hysteresis. When the system's symmetry if broken we utilize numerical techniques to build the bifurcation diagram of the system, uncovering more complicated collective dynamics. The asymmetry breaks a pitchfork bifurcation corresponding to the onset of synchronization. In fact, when symmetry is broken we find that the onset of synchronization always occurs via a Hopf bifurcation. We also find that, instead of transitioning directly to a traveling wave state, the broken symmetry leads to the emergence of a thin region of partial synchronization before the traveling waves state. and such that the loss of stability of the incoherent state always comes via a Hopf bifurcation. Finally, an additional bistable state is observed between to different partially synchronized states representing stronger and weaker synchronization.

Publisher URL: http://arxiv.org/abs/1805.00303

DOI: arXiv:1805.00303v2

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