Cooperative phenomena observed in a globally coupled phase-locked loop system

The authors describe cooperative phenomena observed in a globally coupled 8-phase-locked loop (PLL) system in both experiments and simulations by using SPICE. In the present system, each PLL is connected through the coupler that averages the all PLLs' output signals. When the frequency modulated external force is applied, three phases are observed such as (1) the coherent phase in which all subsystems are synchronized, (2) the turbulent phase in which all subsystems oscillate independently, and (3) the partially synchronized phase in which creation of clusters and chaotic itineracy due to the reconstruction of clusters occur. In the partially synchronized phase, the time series of voltage difference between arbitrary two subsystems shows an intermittency. From the statistical analysis for this intermittency it is found that the distribution functions of burst amplitude and laminar duration time show the -1 power law and -1.5 power law, respectively. These properties agree with those of the on-off intermittency. We also investigate the variation of number of cluster in the partially synchronized phase. The distribution function of the duration time of one-cluster state, namely the residence time in the state in which all subsystems are synchronized with each other, is also expressed as the -1.5 power law.