Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results
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In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the corresponding Ott–Antonsen (OA)-reduced dynamics in the special case of a Lorentzian distribution. Besides elucidating the effects of delay on the nature of bifurcation, we show that the approach due to Ott and Antonsen, although an ansatz, gives an amplitude dynamics of the unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. Further more, quite interestingly and remarkably, we show that close to the bifurcation, the unstable manifold derived from the kinetic equation has the same form as the OA manifold, implying thereby that the OA-ansatz form follows also as a result of the unstable manifold expansion. We illustrate our results by showing how delay can affect dramatically the bifurcation of a bimodal distribution.
KeywordsNonlinear dynamics and chaos Synchronization Coupled oscillators Bifurcation analysis
This work was initiated while DM was affiliated to Laboratory J.A. Dieudonné, Université Côte d’Azur, Nice, France and was finalized with DM being affiliated to Los Alamos National Laboratory (LANL). DM gratefully acknowledges the support of the U.S. Department of Energy through the LANL/LDRD Program and the Center for Non Linear Studies, LANL. The paper was written up during the visit of DM and SG to the International Centre for Theoretical Physics – South American Institute for Fundamental Research, São Paulo, Brazil in May 2018 and during SG’s extended stay at the Universidade Federal de São Carlos and the Centro de Pesquisa em Óptica e Fotônica, Sào Carlos, Brazil during June 2018. The authors thank these institutions for warm hospitality and financial support.
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