Interplay between couplings and common noise in phase synchronization: disagreement between global analysis and local stability characterization

We consider two coupled phase oscillators in the presence of proportional ("common") and independent white noises. The global synchronization properties of the system are analytically studied via the Fokker-Planck equation. When the "effective coupling" is big compared to the common noises, the former favors and the latter hinder synchronization. On the contrary, when the coupling is small compared to the proportional noises, we find that the latter induce synchronization, optimally when their intensities are big and in the n:m synchronization ratio. Furthermore, in such case a small value of the coupling is better for synchronization. Finally, we show that synchronization, which is a global property, must not be studied via local stability such as with Lyapunov exponents.