Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators

Seung-Yeal Ha; Jaeseung Lee; Zhuchun Li; Seung-Yeal Ha; Jaeseung Lee; Zhuchun Li

doi:10.3934/nhm.2017001

Networks and Heterogeneous Media

2017, Volume 12, Issue 1: 1-24. doi: 10.3934/nhm.2017001

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Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea
2.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
3.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received: 01 January 2016 Revised: 01 November 2016
Fund Project: The work of S.-Y. Ha was supported by the National Research Foundation of Korea (NRF2014R1A2A205002096), and the work of Z. Li was supported by the National Natural Sci ence Foundation of China Grant 11401135 and the Fundamental Research Funds for the Central Universities of China (HIT.BRETIII.201501 and HIT.PIRS.201610).
34C15, 34D06, 92D25

We study the emergence of local exponential synchronization in an ensemble of generalized heterogeneous Kuramoto oscillators with different intrinsic dynamics. In the classic Kuramoto model, intrinsic dynamics are given by the Kronecker flow with constant natural frequencies. We generalize the constant natural frequencies to smooth functions that depend on the state and time so that it can describe a more realistic situation arising from neuroscience. In this setting, the ensemble of generalized Kuramoto oscillators loses its synchronization even when the coupling strength is large. This leads to the study of a concept of "relaxed" synchronization, which is called "practical synchronization" in literature. In this new concept of "weak" synchronization, the phase diameter of the entire ensemble is uniformly bounded by some constant inversely proportional to the coupling strength. We focus on the complete synchronizability of a subensemble consisting of generalized Kuramoto oscillators with the same intrinsic dynamics; moreover, we provide several sufficient frameworks leading to local exponential synchronization of each homogeneous subensemble, although the whole ensemble is not fully synchronized. This is a generalization of an earlier analytical result regarding practical synchronization. We also provide several numerical simulations and compare them with analytical results.
- Kuramoto oscillators,
- intrinsic dynamics,
- interacting groups
Citation: Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators[J]. Networks and Heterogeneous Media, 2017, 12(1): 1-24. doi: 10.3934/nhm.2017001

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Abstract

We study the emergence of local exponential synchronization in an ensemble of generalized heterogeneous Kuramoto oscillators with different intrinsic dynamics. In the classic Kuramoto model, intrinsic dynamics are given by the Kronecker flow with constant natural frequencies. We generalize the constant natural frequencies to smooth functions that depend on the state and time so that it can describe a more realistic situation arising from neuroscience. In this setting, the ensemble of generalized Kuramoto oscillators loses its synchronization even when the coupling strength is large. This leads to the study of a concept of "relaxed" synchronization, which is called "practical synchronization" in literature. In this new concept of "weak" synchronization, the phase diameter of the entire ensemble is uniformly bounded by some constant inversely proportional to the coupling strength. We focus on the complete synchronizability of a subensemble consisting of generalized Kuramoto oscillators with the same intrinsic dynamics; moreover, we provide several sufficient frameworks leading to local exponential synchronization of each homogeneous subensemble, although the whole ensemble is not fully synchronized. This is a generalization of an earlier analytical result regarding practical synchronization. We also provide several numerical simulations and compare them with analytical results.

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