# 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: $R(t,L=100)$.

## References

- Pikovsky, A.; Kurths, J.; Rosenblum, M.; Kurths, J. Synchronization: A Universal Concept in Nonlinear Sciences; Cambridge Nonlinear Science Series; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Acebrón, J.A.; Bonilla, L.L.; Vicente, C.J.P.; Ritort, F.; Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys.
**2005**, 77, 137. [Google Scholar] [CrossRef] [Green Version] - Arenas, A.; Díaz-Guilera, A.; Kurths, J.; Moreno, Y.; Zhou, C. Synchronization in complex networks. Phys. Rep.
**2008**, 469, 93–153. [Google Scholar] [CrossRef] [Green Version] - Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence. In Proceedings of the International Symposium on Mathematical Problems in Theoretical Physics, Kyoto, Japan, 23–29 January 1975. [Google Scholar]
- Ott, E.; Antonsen, T.M. Low dimensional behavior of large systems of globally coupled oscillators. Chaos
**2008**, 18, 037113. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hong, H.; Chaté, H.; Park, H.; Tang, L.H. Entrainment transition in populations of random frequency oscillators. Phys. Rev. Lett.
**2007**, 99, 184101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Choi, C.; Ha, M.; Kahng, B. Extended finite-size scaling of synchronized coupled oscillators. Phys. Rev.
**2013**, 88, 032126. [Google Scholar] [CrossRef] [Green Version] - Sakaguchi, H.; Shinomoto, S.; Kuramoto, Y. Local and grobal self-entrainments in oscillator lattices. Prog. Theor. Phys.
**1987**, 77, 1005. [Google Scholar] [CrossRef] [Green Version] - Hong, H.; Park, H.; Choi, M. Collective synchronization in spatially extended systems of coupled oscillators with random frequencies. Phys. Rev. E
**2005**, 72, 036217. [Google Scholar] [CrossRef] [Green Version] - Juhász, R.; Kelling, J.; Ódor, G. Critical dynamics of the Kuramoto model on sparse random networks. J. Stat. Mech. Theory Exp.
**2019**, 2019, 053403. [Google Scholar] [CrossRef] [Green Version] - Filatrella, G.; Nielsen, A.H.; Pedersen, N.F. Analysis of a power grid using a Kuramoto-like model. Eur. Phys. J. B
**2008**, 61, 485–491. [Google Scholar] [CrossRef] [Green Version] - Tanaka, H.A.; Lichtenberg, A.J.; Oishi, S. First order phase transition resulting from finite inertia in coupled oscillator systems. Phys. Rev. Lett.
**1997**, 78, 2104–2107. [Google Scholar] [CrossRef] - Ódor, G.; Hartmann, B. Heterogeneity effects in power grid network models. Phys. Rev. E
**2018**, 98, 022305. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Villegas, P.; Moretti, P.; Munoz, M.A. Frustrated hierarchical synchronization and emergent complexity in the human connectome network. Sci. Rep.
**2014**, 4, 5990. [Google Scholar] [CrossRef] [Green Version] - Millán, A.P.; Torres, J.J.; Bianconi, G. Complex network geometry and frustrated synchronization. Sci. Rep.
**2018**, 8, 9910. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ódor, G.; Deng, S.; Hartmann, B.; Kelling, J. Synchronization dynamics on power grids in Europe and the United States. Phys. Rev. E
**2022**, 106, 034311. [Google Scholar] [CrossRef] - Mermin, N.D.; Wagner, H. Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models. Phys. Rev. Lett.
**1966**, 17, 1133–1136. [Google Scholar] [CrossRef] - Ódor, G.; Hartmann, B. Power-law distributions of dynamic cascade failures in power-grid models. Entropy
**2020**, 22, 666. [Google Scholar] [CrossRef] - Cardy, J.L. One-dimensional models with 1/r2interactions. J. Phys. A Math. Gen.
**1981**, 14, 1407–1415. [Google Scholar] [CrossRef] - Janssen, H.K.; Müller, M.; Stenull, O. Generalized epidemic process and tricritical dynamic percolation. Phys. Rev. E
**2004**, 70, 026114. [Google Scholar] [CrossRef] [Green Version] - Chan, W.; Ghanbarnejad, F.; Grassberger, P. Avalanche outbreaks emerging in cooperative contagions. Nat. Phys.
**2015**, 11, 936–940. [Google Scholar] - Gómez-Gardeñes, J.; Gómez, S.; Arenas, A.; Moreno, Y. Explosive Synchronization Transitions in Scale-Free Networks. Phys. Rev. Lett.
**2011**, 106, 128701. [Google Scholar] [CrossRef] [Green Version] - Coutinho, B.C.; Goltsev, A.V.; Dorogovtsev, S.N.; Mendes, J.F.F. Kuramoto model with frequency-degree correlations on complex networks. Phys. Rev. E
**2013**, 87, 032106. [Google Scholar] [CrossRef] - Ódor, G.; de Simoni, B. Heterogeneous excitable systems exhibit Griffiths phases below hybrid phase transitions. Phys. Rev. Res.
**2021**, 3, 013106. [Google Scholar] [CrossRef] - Grainger, J.J.; Stevenson, W.D. Power System Analysis; McGraw-Hill: New York, NY, USA, 1994. [Google Scholar]
- Ahnert, K.; Mulansky, M. Boost::odeint. Available online: https://odeint.com (accessed on 1 December 2022).
- Jeffrey, K.; Deng, S.; Barancsuk, L.; Hartmann, B.; Ódor, G. Solving the Kuramoto Equation by GPUs; Centre for Energy Research: Budapest, Hungary, to be published.
- Ódor, G.; Deco, G.; Kelling, J. Differences in the critical dynamics underlying the human and fruit-fly connectome. Phys. Rev. Res.
**2022**, 4, 023057. [Google Scholar] [CrossRef] - Ódor, G.; Kelling, J.; Deco, G. The effect of noise on the synchronization dynamics of the Kuramoto model on a large human connectome graph. J. Neurocomput.
**2021**, 461, 696–704. [Google Scholar] [CrossRef]

**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.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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