Phase Locking of Linear Oscillators with Individual Parameters

A recent paper has shown that linear oscillators can be synchronised only if their parameters are exactly the same. The main reason for this sensitivity lies in the fact that the oscillator network loses energy whenever the oscillators do not follow precisely the same output trajectory. This paper deals with the question how to extend oscillators to make them synchronisable in a practical sense. The oscillators are equipped with an energy source that replaces the energy lost during the synchronisation. It is shown that a power supply rate exists such that oscillators with different eigenfrequencies can phase lock, which means that they follow the same sinusoidal trajectory with some phase gap. For two coupled oscillators explicit relations for the frequency and the phase shift of the synchronous behaviour are derived.