The instability of the incoherent state under Lévy noise: A unified Lévy–Penrose criterion for Kuramoto-type models

The instability of the incoherent state is a central problem in Kuramoto oscillator systems, particularly under non-Gaussian perturbations such as α-stable Lévy noise. In this paper, we study the instability of the incoherent state in a nonlocal Kuramoto–Sakaguchi model driven by α-stable Lévy noise. By extending the Penrose stability approach to this stochastic setting, we establish a unified Lévy–Penrose criterion and obtain a theoretical spectral reference coupling strength by examining the neutral spectral boundary along the imaginary axis, λ=iΩ. The corresponding Penrose curves are used to show how Lévy noise and the intrinsic frequency distribution modify the spectral structure. Numerical results indicate that this reference coupling strength is closely related to the coupling range over which the structural reorganization of the phase distribution becomes clearly visible, rather than to an abrupt transition to complete synchronization.