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# Communications in Mathematical Sciences

## Volume 11 (2013)

### Number 2

### Exponential synchronization of finite-dimensional Kuramoto model at critical coupling strength

Pages: 385 – 401

DOI: http://dx.doi.org/10.4310/CMS.2013.v11.n2.a3

#### Authors

#### Abstract

We discuss the exponential synchronization for an ensemble of Kuramoto oscillators at the critical coupling strength, which is the diameter of the set consisting of natural frequencies. When the number of distinct natural frequencies is greater than two and the initial phases are strictly confined in an interval of length π∕2, we show that the initial configuration evolves toward a phase-locked state at least exponentially fast. This fast convergence toward the phase-locked state is markedly different from an ensemble of Kuramoto oscillators with only two distinct natural frequencies. For this, we derive a Gronwall inequality for the frequency diameter to obtain complete synchronization. We also compare our analytical results with numerical simulation results.

#### Keywords

critical coupling strength, exponential synchronization, Kuramoto model, natural frequency

#### 2010 Mathematics Subject Classification

74A25, 76N10, 92D25