Collective phenomena in large populations of globally coupled relaxation oscillators.

Abstract

We present a study of a large pool of globally coupled relaxation oscillators. The reaction of the pool to the presence of a modulating external field is discussed. The coupling is assumed homogeneous and linear. Randomly distributed internal frequencies introduce a disordering element that, due to the coupling, can result in oscillator quiescence. Self-synchronization is shown to be absent in this system. However, this is entirely due to the linear coupling. For identical oscillators the basic state is incoherent and marginally stable in an extended region of parameter space. With modulation on the levels, the average rotation number as function of the external frequency lies on a devil's staircase, as for a single oscillator. However, the locked regions shrink with increasing coupling. This has some important consequences for the critical lines of the averaged system. In the case of modulation on the frequency (no damping), the average rotation number is still independent of the external signal, as for a single oscillator. The real surprise lies in the resulting distribution of individual frequencies or rotation numbers. No matter how the external field is applied, this distribution is forced into a devil's staircase exhibiting a critical point, which in the case of modulation on the lower level is a transition from quasiperiodicity to chaos and otherwise to completeness.