Abstract
We consider two finite-dimensional models of the phase oscillators in the case of star configuration of coupling. Both systems of equations are reduced to a nonlinearly coupled system of pendulum equations. We prove that the transition from synchronous to asynchronous oscillations occurs via bifurcation of saddle-node equilibrium. In this connection the asynchronous regime can be partially synchronous rotations. We find that the reverse transition from asynchronous to synchronous regime occurs via bifurcation of homoclinic orbit both of the saddle equilibrium point and of the saddle periodic orbit. In the case of homoclinic loop of the saddle point the synchrony appears only from asynchronous mode without partially synchronized rotations. In the case of the homoclinic curve of the saddle periodic orbit the system undergoes a chaotic rotation regime which results in a random return to synchrony.
M.I. Rabinovich is one of the pioneers in the field of synchronization in complex networks of oscillators [1]. We dedicate the present piece of work to Misha’s 75th birthday.
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Acknowledgements
This work was supported by the RSF (Project No. 14-12-00811) (Sections 1, 2) and by the RFBR (project 15-01-08776) (Section 3).
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Belykh, V.N., Bolotov, M.I., Osipov, G.V. (2017). Regular and Chaotic Transition to Synchrony in a Star Configuration of Phase Oscillators. In: Aranson, I., Pikovsky, A., Rulkov, N., Tsimring, L. (eds) Advances in Dynamics, Patterns, Cognition. Nonlinear Systems and Complexity, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-53673-6_7
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