Synchronization for the Rough Kuramoto Model

Abstract

We study the local synchronization of phases and frequencies for the Kuramoto model driven by rough noise. In particular, we prove exponential convergence towards synchronization and we give the explicit rate of convergence and quantify the size of the random basin of attraction. Furthermore, we show that the long time behavior of the system is determined by the evolution of phases' mean. Our result relies on the use of a Lyapunov function, capable of overriding the particular structure of the noise, taking in account only its intensity. Finally, we illustrate our analytical results and possible extensions with the help of numerical simulations.

Figures (7)

• Figure 1: Simulation of the trajectory $\theta(t) = (\theta_1(t), \theta_2(t), \theta_3(t))$ for the all-to-all coupling case, with initial data satisfying Corollary \ref{['Cor_man']}, shown from two different perspectives.
• Figure 2: Plots illustrating phase synchronization in accordance with our theoretical results. The left panel shows the case of all-to-all coupling, while the right panel corresponds to a general symmetric adjacency matrix.
• Figure 3: Plots showing that synchronization occurs even when our hypotheses are violated. The left panel corresponds to the case of all-to-all coupling, while the right panel shows the case of a general symmetric adjacency matrix.
• Figure 4: Left: splitting observed in the all-to-all coupling configuration for $\sigma \gg 1$. Right: synchronization in a disconnected graph, arising from the connected structure of the noise function.
• Figure 5: Left: synchronization for $G_i(\theta) = \sin(\theta_i)$ with identical noise. Right: desynchronization when the components of $\textbf{W}$ are non-identical.

Theorems & Definitions (26)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.2
• proof
• Definition 2.3
• Theorem 2.4
• proof
• Remark 2.4
• Theorem 2.5