Given the dynamics of both oscillators:
we denote the variable that represents time difference
where is the moment in time when crosses the zero line from positive to negative, and is the moment in time when crosses the zero line from positive to negative. is the frequency of both oscillators. For this project we assume that .
We define set , and claim that it is attractive. To support this claim, we need to prove that all trajectories that start close to the set satisfy:
For any initial condition of , control algorithm will speed up the slave physical system to a period of time that is proportional to the value of . The system speed-up, in turn, guarantees the decrease in phase difference. Consequently, it guarantees that the next value of the x will less than the previous. In other words, all the trajectories strictly approach 0.