Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators

Seung-Yeal Ha; Jaeseung Lee; Zhuchun Li

doi:10.3934/nhm.2017001

Article Contents

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

This issue Previous Article Next Article Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling

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: December 31, 2015

Revised: October 31, 2016

Published: May 2017

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).

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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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Mathematics Subject Classification: 34C15, 34D06, 92D25.

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Figure 1. Plot of phases versus time

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Figure 2. Bipartite network

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Figure 3. All-to-all network

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Figure 4. Mixture of three homogeneous ensembles on an all-toall network

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