Exponential synchronization of the Kuramoto model with inertia and frustration under locally coupled networks

Tingting Zhu; Xiongtao Zhang

doi:10.3934/dcdsb.2025135

Article Contents

2026, Volume 32: 355-383. Doi: 10.3934/dcdsb.2025135

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Exponential synchronization of the Kuramoto model with inertia and frustration under locally coupled networks

Tingting Zhu^{1, 2,} and
Xiongtao Zhang^3, ,

1.
School of Mathematics and Statistics, Hefei University, Hefei, China
2.
Key Laboratory of Applied Mathematics and Artificial Intelligence Mechanism, Hefei University, Hefei, China
3.
School of Mathematics and Statistics, Wuhan University, Wuhan, China

^*Corresponding author: Xiongtao Zhang
^*Corresponding author: Xiongtao Zhang

Received: April 2025

Revised: June 2025

Early access: August 29, 2025

Published online: August 29, 2025

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Abstract

We study the collective synchronized behavior of the Kuramoto model with inertia and frustration effects on a connected and symmetric network. We aim to establish sufficient frameworks for achieving complete frequency synchronization, taking into account initial configuration, small inertia and frustration, and large coupling strength. More precisely, we first demonstrate that the phase diameter will be uniformly bounded by a small value after a finite time. Then, we prove that the frequency diameter exhibits exponential decay to zero. Our approach relies on a careful construction of energy functionals, which effectively control the dissipation of phase and frequency diameters.

Keywords:

Mathematics Subject Classification: Primary: 34D05, 34D06; Secondary: 34C15, 92B25.

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Figure 1. Complete synchronization with $ m = 0.000001, K = 1000, \alpha = 0.0001 $

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Figure 2. Dynamics of $ \log D(\omega(t)) $ with $ m = 0.000001, K = 1000, \alpha = 0.0001 $

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Figure 3. Complete synchronization with $ m = 0.00001, K = 100, \alpha = 0.01 $

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Table 1. The slope for $ \log D(\omega(t)) $ near $ t = 0.03 $ and $ 0.04 $

Time	Slope	Time	Slope
0.030000	-333.4441	0.039990	-333.4441
0.030001	-333.4441	0.039991	-333.4441
0.030002	-333.4441	0.039992	-333.4441
0.030003	-333.4441	0.039993	-333.4441
0.030004	-333.4441	0.039994	-333.4441
0.030005	-333.4441	0.039995	-333.4441
0.030006	-333.4441	0.039996	-333.4441
0.030007	-333.4441	0.039997	-333.4441
0.030008	-333.4441	0.039998	-333.4441
0.030009	-333.4441	0.039999	-333.4441

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