# Synchronization Transition of the Second-Order Kuramoto Model on Lattices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Methods

#### 2.1. The Second-Order Kuramoto Model

#### 2.2. Linear Approximation for the Frequency Entrainment

## 3. Synchronization Transition in 2D

#### 3.1. Frequency Entrainment Phase Transition

#### 3.2. Phase-Order Parameter Transition

## 4. Synchronization Transition in 3D

#### 4.1. Frequency Entrainment Phase Transition

#### 4.2. Phase-Order Parameter Transition

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**The frequency spread in 2D at $\alpha =0.4$ for different K values, shown by the legends for $L=2000$, with disordered initial conditions. The dashed line marks a numerical fit at the critical point at ${K}_{c}=9.5\left(5\right)$ with ${t}^{-0.96\left(5\right)}$.

**Figure A2.**Steady state Kuramoto-order parameter (black ‘+’) in 3D and its variance (red ‘*’) at $\alpha =3$ for different ${K}^{\prime}$ values for $L=100$. Inset: $R(t,L=100)$.

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**Figure 1.**The frequency spread in 2D at $\alpha =3$ for different K values, shown by the legends, for $L=2000$, in the case of ordered initial conditions. The dashed line marks a numerical fit at the critical point ${K}_{c}=3.4\left(1\right)$ with ${t}^{-d/2}$. Inset: the finite size scaling of the frequency entrainment transition point ${K}_{c}$ for various system sizes in 2D (black asterisks) and 3D (red boxes), for $\alpha =3$ and under ordered initial conditions. One can see a logarithmic growth in 2D and a convergence to ${K}_{c}=1.15\left(5\right)$ constant value in 3D.

**Figure 2.**The frequency spread in 2D at $\alpha =0.4$ for different K values, shown by the legends, for $L=2000$, using ordered initial conditions. The dashed line marks a numerical fit at the critical point ${K}_{c}=3.5\left(5\right)$ with ${t}^{-1.03\left(3\right)}$. Inset: Steady state values obtained by starting from ordered (black bullets) and disordered (red boxes) initial conditions.

**Figure 3.**The frequency spread in 2D at $\alpha =3$ for different K values, shown by the legends, for $L=2000$, in the case of disordered initial conditions. The dashed line marks a numerical fit at the critical point at ${K}_{c}=8.0\left(5\right)$ with ${t}^{-1.09\left(5\right)}$. Inset: Part of the hysteresis loop of R in 2D obtained by ordered (black bullets) and disordered (red boxes) initial conditions for $\alpha =3$ and $L=200$.

**Figure 4.**Steady -state Kuramoto order parameter (black dots) in 2D and its variance (red squares) at $\alpha =3$ at different K values for $L=200$. Inset: $R(t,L=200)$ for $K=$ 1, 2, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 20, 25, 35, 45 (bottom to top curves).

**Figure 5.**Finite-size behavior of R in 2D for $\alpha =3$ and ordered initial conditions shows a crossover. Inset: finite-size scaling of ${K}_{c}^{\prime}$ as estimated by the half values of R (black boxes) and by the $\sigma \left(R\right)$ peaks (red bullets) exhibit a linear growth.

**Figure 6.**The frequency spread in 3D at $\alpha =3$ for $K=$ 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.05, 1.1, 2 (top-to-bottom curves) for $L=200$ linear sized lattices and phase-ordered initial conditions. The dashed line marks a numerical fit at the critical point ${K}_{c}=1.02\left(2\right)$ with ${t}^{-d/2}$. Inset: Steady state values obtained by starting from ordered (black bullets) and disordered (red boxes) initial conditions.

**Figure 7.**The frequency spread in 3D at $\alpha =3$ for different K values, shown by the legends, for $L=200$ and disordered initial conditions. The dashed line marks a numerical fit at the critical point $K={K}_{c}\simeq 7$ with ${t}^{-1.8\left(1\right)}$.

**Figure 8.**Finite-size behavior of R in 3D for $\alpha =3$ and the ordered initial conditions, shows a crossover. Inset: finite-size scaling of ${K}_{c}^{\prime}$ as estimated by the half values of R (black bullets) as well as by the $\sigma \left(R\right)$ peaks (red boxes) exhibit a power-law growth.

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Ódor, G.; Deng, S. Synchronization Transition of the Second-Order Kuramoto Model on Lattices. *Entropy* **2023**, *25*, 164.
https://doi.org/10.3390/e25010164

**AMA Style**

Ódor G, Deng S. Synchronization Transition of the Second-Order Kuramoto Model on Lattices. *Entropy*. 2023; 25(1):164.
https://doi.org/10.3390/e25010164

**Chicago/Turabian Style**

Ódor, Géza, and Shengfeng Deng. 2023. "Synchronization Transition of the Second-Order Kuramoto Model on Lattices" *Entropy* 25, no. 1: 164.
https://doi.org/10.3390/e25010164