In previous chapters we considered synchronization in purely deterministic systems, neglecting all irregularities and fluctuations. Here we discuss how the latter effects can be incorporated in the picture of phase locking. We start with a discussion of the effect of noise on autonomous self-sustained oscillations. We show that noise causes phase diffusion, thus spoiling perfect time-periodicity. Next, we consider synchronization by an external periodic force in the presence of noise. Finally, we discuss mutual synchronization of two noisy oscillators.

Self-sustained oscillator in the presence of noise

No oscillator is perfectly periodic: all clocks have to be adjusted from time to time, some even rather often. There are many factors causing irregularity of self-sustained oscillators, for simplicity we will call them all noise. A detailed analysis of noisy oscillators must include a thorough mathematical description of the problem, where fluctuations of different nature (e.g., technical, thermal) should be taken into account. This has been done for different types of oscillators (see, e.g., [Malakhov 1968]); here we want to discuss the basic phenomena only.

As the first model, we consider a self-sustained oscillator subject to a noisy external force. In revising the basic equations of forced oscillators of Chapter 7, one can see that only the approximation of phase dynamics is valid in the case of a fluctuating force as well, since we do not assume any regularity of the force in the derivation of Eq. (7.15).