A Universal Concept in Nonlinear Sciences
By Arkady Pikovsky
By Michael Rosenblum
By Jürgen Kurths
Cambridge Nonlinear Science Series (No. 12)
Publisher: Cambridge University Press
Print Publication Year: 2001
Online Publication Date:July 2010
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511755743.013
Subjects: Nonlinear Science and Fluid Dynamics
The effect of mutual synchronization of two coupled oscillators, described in Chapter 8, can be generalized to more complex situations. One way to do this was described in Chapter 11, where we considered lattices of oscillators with nearest-neighbor coupling. However, often oscillators do not form a regular lattice, and, moreover, interact not only with neighbors but also with many other oscillators. Studies of three and four oscillators with all-to-all coupling give a rather complex and inexhaustible picture (see, e.g., [Tass and Haken 1996; Tass 1997]). The situation becomes simpler if the interaction is homogeneous, i.e., all the pairs of oscillators are equally coupled. Moreover, if the number of interacting oscillators is very large, one can consider the thermodynamic limit where the number of elements in the ensemble tends to infinity. Now, when the oscillators are not ordered in space, one usually speaks of an ensemble or a population of coupled oscillators. An analogy to statistical mechanics is extensively exploited, so it is not surprising that the synchronization transition appears as a nonequilibrium phase transition in the ensemble.
We start with the description of the Kuramoto transition in a population of phase oscillators. An ensemble of noisy oscillators is then discussed. Different generalizations of these basic models (e.g., populations of chaotic oscillators) are described in Section 12.3.