Asynchronous and synchronous states in the heterogeneous Kuramoto model

• Applicant: Dr. Yagmur Kati
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics ; Theoretical Condensed Matter Physics
• Term: from 2022 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 506198590

• Final Report Year: 2025

Final Report Abstract

Numerous natural and technical systems rely on the interaction of individual units that influence one another, such as neurons in the brain, heart cells, or generators in power grids. Often, these systems synchronize, meaning all units behave in a coordinated way. The Kuramoto model is a well-known mathematical model used to study how synchronization happens. While synchronized states have been widely studied, less is known about asynchronous states, where the units stay out of sync but still interact. This project focused on understanding such asynchronous states in complex systems where the properties of individual units and their connections vary randomly. In the first part of the project, we studied systems without inertia and developed the iterative mean-field (IMF) method. This method makes it possible to understand the behavior of large networks by analyzing a single unit in a statistically consistent way. We showed that this approach gives reliable results even in small systems and for different kinds of randomness. In the second part of the project, we extended the method to systems with inertia, which are more realistic in many physical and biological settings. We found that at intermediate values of inertia, the fluctuations in the system become especially uncorrelated, and the system behaves almost like white noise. This state also shows the highest level of dynamical complexity. Our findings help explain how asynchronous behavior works in systems where synchronization is not always the goal. This is important in areas like neuroscience or power systems, where synchronization is not always desirable and can sometimes lead to problems. The approach we developed could also be applied in other areas where large, noisy systems interact, such as in modeling brain activity, studying the stability of power grids, understanding collective motion in animal groups, or analyzing patterns in climate systems.

Publications

• Iterative Mean Field Method and Network Dynamics of Heterogeneous Kuramoto Model
  Y. Kati, J. Ranft & B. Lindner
  
• Self-consistent autocorrelation of a disordered Kuramoto model in the asynchronous state. Physical Review E, 110(5).
  Kati, Yagmur; Ranft, Jonas & Lindner, Benjamin
  
• Effects of inertia on the asynchronous state of a disordered Kuramoto model. Physical Review E, 112(4).
  Kati, Yagmur; Toenjes, Ralf & Lindner, Benjamin
  
• Power Spectrum Analysis of Asynchronous States in Kuramoto Oscillators with Inertia
  Y. Kati
  
• Supporting Code for: Effects of inertia on the asynchronous state of a disordered Kuramoto model
  Y. Kati