Nonlinear synchronization of coupled oscillators: The polynomial case

This paper presents a feedback method to achieve synchronization of coupled identical oscillators which are characterized by polynomial vector fields. Here, synchronization means asymptotic coincidence of the states of all the systems. Even though their models are identical, the state trajectories of the identical systems are different because of different initial conditions. Unlike other approaches where just a linear damping term is added to each system in order to achieve synchronization, we design nonlinear coupling functions between the subsystems in such a way that stability of the error dynamics between any two models results. To do that, a certain dissipation inequality and sum of squares as a computational tool are used. Finally, two examples are presented to illustrate the proposed method.