Stability of uniformly incoherent state in the D-dimensional generalized Kuramoto model

In this paper, we are devoted to theoretically analyzing the stability of the completely incoherent state in the D-dimensional generalized Kuramoto model within the same one framework, where a completely incoherent state refers to that all agents are uniformly distributed on surface S of the unit sphere in D-dimensional space. By linearizing the continuity equation of the model in its thermodynamic limit, we obtain the characteristic equation that governs the linear stability of the uniformly incoherent state for arbitrary dimension with D≥2. Moreover, we show that all the stability information regarding the complete incoherence can be successfully retrieved from the reduced system via high-dimensional Ott-Antonsen ansatz for the D-dimensional generalized Kuramoto model. For a Gaussian ensemble of natural rotations, we demonstrate that the characteristic equation can be explicitly simplified for both even and odd D. In particular, via the simplified characteristic equation, we verify theoretically that the critical coupling strength for the instability of the uniformly incoherent state is always retained at zero for all odd D≥3. Our study provides a detailed recipe for the stability analysis of complete incoherence in the D-dimensional generalized Kuramoto model, which is potential for identifying the onset of phase transition to synchrony in systems of interacting high-dimensional heterogeneous agents.