Frequency Locking For Some Kind Of Nonlinear Systems

This paper is devoted to the study of the asymptotic dynamics of a specific system of coupled oscillators in the deterministic case and in the stochastic case. In the deterministic case, we assume the nonlinear coefficient increases at most linearly while the linear matrix is under certain conditions. So we get a solution of the system uniformly bounded by the application of Lypanov functions. By defining the rotation number as the limit of frequency of the oscillator, we can prove that the system has rotation numbers in every direction. In the stochastic case, we add stochastic disturbance to the deterministic system. It is shown that we can address a positive recurrent diffusion process by winding the solution of the stochastic system on a cylinder along a special direction. So the rotation number exists in every direction no matter what the diffusion matrix is.