Theoretical and Computational Studies to Understand the Behavior of the Kuramoto Model in Different Network Topologies
Vishrut Kinikar
21/07/2026
The localization of the Kuramoto model in various real-world systems such as Josephson junctions and ring oscillator circuits has emphasized the need to understand the behavior of the Kuramoto model in localized network topologies. This study investigates the evolution of the order parameter of a dynamical system in the ring and chain topologies, and records their respective final average order parameters for increasing numbers of oscillators. Two cases were tested: identical and heterogeneous frequencies. In both cases, the values for the final average order parameter and the irregular oscillatory motion of individual nodes converged in the thermodynamic limit for both topologies. Then, the case of heterogeneous frequencies was tested for small oscillator amounts. Divergence in the values of the final average order parameter of the ring and chain were observed when compared. Interestingly, no clear relationship was observed between topological differences and oscillatory fluctuations. Lastly, the synchronization transitions of the all-to-all Kuramoto model and ring topology were compared in the case of heterogeneous frequencies. The result obtained was that while the standard all-to-all model had a definitive critical coupling constant where a continuous phase transition occurred, the ring topology exhibited no such transition due to structural differences. This demonstrated the role of network topology in the synchronization dynamics of an oscillator system. Understanding these effects of topology on dynamic system behavior in a synthesized manner is an endeavor that this paper aims to make contributions towards.