2023/09/01 by Alex Arenas, Àlex Arenas, Antonio Garijo +2 · 1 citation
Computer Science · Engineering · #Nonlinear Dynamics and Pattern Formation #Slime Mold and Myxomycetes Research
paper · pdf · doi:10.1063/5.0161977
openalex publication_date 2023/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider a system of n coupled oscillators described by the Kuramoto model with the dynamics given by θ˙=ω+Kf(θ). In this system, an equilibrium solution θ∗ is considered stable when ω+Kf(θ∗)=0, and the Jacobian matrix Df(θ∗) has a simple eigenvalue of zero, indicating the presence of a direction in which the oscillators can adjust their phases. Additionally, the remaining eigenvalues of Df(θ∗) are negative, indicating stability in orthogonal directions. A crucial constraint imposed on the equilibrium solution is that |Γ(θ∗)|≤π, where |Γ(θ∗)| represents the length of the shortest arc on the unit circle that contains the equilibrium solution θ∗. We provide a proof that there exists a unique solution satisfying the aforementioned stability criteria. This analysis enhances our understanding of the stability and uniqueness of these solutions, offering valuable insights into the dynamics of coupled oscillators in this system.