In this chapter we describe synchronization by external forces; we shall discuss effects other than those presented in Chapters 7 and 10. The content is not homogeneous: different types of systems and different types of forces are discussed here. Nevertheless, it is possible to state a common property of all situations: synchronization occurs when the driven system loses its own dynamics and follows those of the external force. In other words, the dynamics of the driven system are synchronized if they are stable with respect to internal perturbations. Quantitatively this is measured by the largest Lyapunov exponent: a negative largest exponent results in synchronization (note that here we are speaking not about a transverse or conditional Lyapunov exponent, but about the “canonical” Lyapunov exponent of a dynamical system). This general rule does not depend on the type of forcing or on the type of system; nevertheless, there are some problem-specific features. Therefore, we consider in the following sections the cases of periodic, noisy, and chaotic forcing separately. In passing, it is interesting to note that we can also interpret phase locking of periodic oscillations by periodic forcing (Chapter 7) as stabilization of the dynamics: a nonsynchronized motion (unforced, or outside the synchronization region) has a zero largest Lyapunov exponent, while in the phase locked state it is negative.

Another important concept we present in this chapter is sensitivity to the perturbation of the forcing. In contrast to sensitivity to initial conditions, which is measured via the Lyapunov exponent, sensitivity to forcing has no universal quantitative characteristics.