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Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

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Abstract

We construct a nontrivial generalization of the paradigmatic Kuramoto model by using an additional coupling term that explicitly breaks its rotational symmetry resulting in a variant of the Winfree model. Consequently, we observe the characteristic features of the phase diagrams of both the Kuramoto model and the Winfree model depending on the degree of the symmetry breaking coupling strength for unimodal frequency distribution. The phase diagrams of both the Kuramoto and the Winfree models resemble each other for symmetric bimodal frequency distribution for a range of the symmetry breaking coupling strength except for region shift and difference in the degree of spread of the macroscopic dynamical states and bistable regions. The dynamical transitions in the bistable states are characterized by an abrupt (first-order) transition in both the forward and reverse traces. For asymmetric bimodal frequency distribution, the onset of bistable regions depends on the degree of the asymmetry. Large degree of the symmetry breaking coupling strength promotes the synchronized stationary state, while a large degree of heterogeneity, proportional to the separation between the two central frequencies, facilitates the spread of the incoherent and standing wave states in the phase diagram for a low strength of the symmetry breaking coupling. We deduce the low-dimensional equations of motion for the complex order parameters using the Ott-Antonsen ansatz for both unimodal and bimodal frequency distributions. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations of the generalized Kuramoto model agree well with the analytical bifurcation curves.

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Data Availability Statement

The data sets on the current study are available from the corresponding author on reasonable request. [Authors’ comment: The homoclinic bifurcation curve is obtained by using XPPAUT software [43].]

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Acknowledgements

The work of V.K.C. is supported by the DST-CRG Project under Grant No. CRG/2020/004353 and DST, New Delhi for computational facilities under the DST-FIST program (SR/FST/PS- 1/2020/135)to the Department of Physics. M.M. thanks the Department of Science and Technology, Government of India, for providing financial support through an INSPIRE Fellowship No. DST/INSPIRE Fellowship/2019/IF190871. S.G. acknowledges support from the Science and Engineering Research Board (SERB), India under SERB-TARE scheme Grant No. TAR/2018/000023 and SERB-MATRICS scheme Grant No. MTR/2019/000560. He also thanks ICTP—The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy for support under its Regular Associateship scheme. DVS is supported by the DST-SERB-CRG Project under Grant No. CRG/2021/000816.

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Authors and Affiliations

  1. Department of Physics, Center for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur, 613 401, India

    M. Manoranjani & V. K. Chandrasekar

  2. Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai, 400005, India

    Shamik Gupta

  3. Department of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram, 695016, India

    D. V. Senthilkumar

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  1. M. Manoranjani
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  3. D. V. Senthilkumar
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  4. V. K. Chandrasekar
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Correspondence to V. K. Chandrasekar.

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Manoranjani, M., Gupta, S., Senthilkumar, D.V. et al. Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling. Eur. Phys. J. Plus 138, 144 (2023). https://doi.org/10.1140/epjp/s13360-023-03760-5

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