Abstract
We analyze the synchronization transition of a globally coupled network of N phase oscillators with inertia (rotators) whose natural frequencies are unimodally or bimodally distributed. In the unimodal case, the system exhibits a discontinuous hysteretic transition from an incoherent to a partially synchronized (PS) state. For sufficiently large inertia, the system reveals the coexistence of a PS state and of a standing wave (SW) solution. In the bimodal case, the hysteretic synchronization transition involves several states. Namely, the system becomes coherent passing through traveling waves (TWs), SWs and finally arriving to a PS regime. The transition to the PS state from the SW occurs always at the same coupling, independently of the system size, while its value increases linearly with the inertia. On the other hand the critical coupling required to observe TWs and SWs increases with N suggesting that in the thermodynamic limit the transition from incoherence to PS will occur without any intermediate states. Finally a linear stability analysis reveals that the system is hysteretic not only at the level of macroscopic indicators, but also microscopically as verified by measuring the maximal Lyapunov exponent.
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Acknowledgments
We would like to thank E.A. Martens, D. Pazó, E. Montbrió, M. Wolfrum for useful discussions. We acknowledge partial financial support from the Italian Ministry of University and Research within the project CRISIS LAB PNR 2011-2013. This work is part of of the activity of the Marie Curie Initial Training Network ‘NETT’ project # 289146 financed by the European Commission. A.T. has also been supported by the A\(^*\)MIDEX grant (No. ANR-11-IDEX-0001-02) funded by the French Government “program Investissements d’Avenir”.
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Olmi, S., Torcini, A. (2016). Dynamics of Fully Coupled Rotators with Unimodal and Bimodal Frequency Distribution. In: Schöll, E., Klapp, S., Hövel, P. (eds) Control of Self-Organizing Nonlinear Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-28028-8_2
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