Partial Exponential Stability Analysis of Slow-fast Systems via Periodic Averaging

This letter presents some new criteria for partial exponential stability of a slow-fast nonlinear system with a fast scalar variable using periodic averaging methods. Unlike classical averaging techniques, we construct an averaged system by averaging over this fast scalar variable instead of the time variable. We show that partial exponential stability of the averaged system implies that of the original one. We then apply the obtained criteria to the study of remote synchronization of Kuramoto-Sakaguchi oscillators coupled by a star network with two peripheral nodes. We show that detuning the natural frequency of the central mediating oscillator increases the robustness of the remote synchronization against phase shifts. This work appears to be the first-known attempt to analytically study phase-unlocked remote synchronization.