Synchronization as a Mode of Self-Organization

Synchronization or entrainment is a key concept to the understanding of selforganization phenomena occurring in the fields of coupled oscillators of the dissipative type. We may even say that Part II is devoted to the consideration of this single mode of motion in various physical situations. Specifically, Chap. 6 is concerned with wave phenomena and pattern formation, which may be viewed as typical synchronization phenomena in distributed systems. In contrast, we shall study in Chap. 7 turbulence in reaction-diffusion systems, which is caused by desynchronization among local oscillators. Chapter 5 deals with self-synchronization phenomena in the discrete populations of oscillators where the way they are distributed in physical space is not important (for reasons stated later). We shall introduce some kind of randonmess by assuming that the oscillators are either different in nature from each other or at best statistically identical. One may then expect phase-transition-like phenomena, characterized by the appearance or disappearance of collective oscillations in the oscillator communities. In describing such a new class of phase transitions. Method I turns out to be very useful. 

In spite of its great potential importance, the subject of mutual entrainment in multi-oscillator systems has been little explored so far. The earlier attempt due to Winfree is based on a phase description (Winfree, 1967). There have been some efforts to make his idea more precise in some respects (Kuramoto, 1975, 1981). Approaches not based on a phase description also exist (Aizawa, 1976 Yamaguchi et al., 1981). The theory presented below is partly in common with Neu s recent work on the populations of oscillators, although the latter does not treat phase-transition-like phenomena (Neu, 1980). There are some phase description approaches on the mutual entrainment of two oscillators (Neu, 1979 b Fuji and Sawada, 1978). This work was partly motivated by Marek an Stuchl s 

Finally, we note that in this chapter, especially in Sects. 5.5 -7, we illustrate how the slaving principle works even when fluctuating forces are present. In this connection, the detailed treatment of the slaving principle including noise, which was recently presented by Haken and Wunderlin (1982), is worth mentioning, although they treat the problem in a physical or mathematical context rather different from the present one. 

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