Stability and synchronization of oscillators: new Lyapunov functions

The analysis of asymptotic stability of nonlinear oscillator is one of the classical problems in the theory of oscillations. Usually, it is solved by exploiting Lyapunov functions having the meaning of the full energy of the system. However, the Barbashin–Krasovskii theorem has to be used along this way, and no estimates can be found for the rate of convergence of the trajectories to equilibria points. In this paper we propose a different Lyapunov function which lacks transparent physical meaning. With this function, both the rate of convergence and the domain of attraction of equilibria points can be estimated. This result also enables an efficient analysis of another problem, synchronization of oscillations of two oscillators. We formulate conditions that guarantee frequency synchronization and, on top of that, phase synchronization. Generalizations to the case of arbitrary number of oscillators are also discussed; solution of this problem is crucial in the analysis of power systems.