Stochastic Motion in Many Degrees of Freedom

摘要

In the first year of the research under the AASERT Grant we have developed a morerigorous approach to studying the synchronization of coupled digital oscillators, the bifurcations in the system, and the transition to chaotic dynamics. We have developed a mapping for a general class of coupled digital oscillators, which we call coupled phase graph (CPG) dynamical systems; coupled digital phase locked loops (DPLL's) is a special case of the CPG dynamics. By enforcing symmetries on the dynamics of these mappings we can prove results about phase locking and have demonstrated the existence of horseshoes (chaotic behavior). We have derived results concerning the existence of period-1 and period-2 orbits in the matched CPG dynamics and have applied these results to the coupled DPLLs. In the second year of the grant the work has been generalized to include large numbers of coupled oscillators, connected in various topological configurations. Rigorous results on the fixed points and their stability has been obtained. The third year of the grant will support work on the effect of noise on the dynamics of digital systems.