In this chapter we describe synchronization of periodic oscillators by a periodic external force. The main effect here is complete locking of the oscillation phase to that of the force, so that the observed oscillation frequency coincides exactly with the frequency of the forcing.

We start our consideration with the case of small forcing. In Section 7.1 we use a perturbation technique based on the phase dynamics approximation. This approach leads to a simple phase equation that can be treated analytically. This equation is, however, nonuniversal, as its form depends on the particular features of the oscillator. Another analytic approach is presented in Section 7.2; here we assume not only that the force is small, but also that the periodic oscillations are weakly nonlinear. This enables us to use a method of averaging and to obtain universal equations depending on a few parameters. Historically, this is the first analytical approach to synchronization going back to the works of Appleton [1922], van der Pol [1927] and Andronov and Vitt [1930a,b]. The averaged equations can be analyzed in full detail, but their applicability is limited: in fact, quantitative predictions are possible only for small-amplitude self-sustained oscillations near the Hopf bifurcation point of their appearance.

Generally, when the forcing is not small and/or the oscillations are strongly nonlinear, we have to rely on the qualitative theory of dynamical systems. The tools used here are the annulus and the circle maps described in Section 7.3. This approach gives a general description up to the transition to chaos, it allows one to find limits of the analytical methods and provides a framework for numerical investigations of particular systems.