We consider a phase-locked loop for the case of an external signal with a stationary fluctuating phase. The problem reduces to the problem of a Brownian particle in a periodic potential driven by “green” noises. We numerically simulate the case in which the random phase is the Ornstain-Uhlenbeck process. The rapid irreversible transition from stationary random motion (a locked state) to a nonstationary one at a high near-constant rate (a running state) is shown to be possible for the case of the massive particle. We found that transition moments change suddenly for small variations of external parameters. We call this phenomenon the “catastrophe”. The numerical results are compared with those obtained by the Krylov-Bogoliubov averaging method. The first approximation of the method is found to be sufficiently accurate if the states coexist and the direct and backward transitions occur frequently enough.

Date Published: 7 May 2003
PDF: 9 pages
Proc. SPIE 5114, Noise in Complex Systems and Stochastic Dynamics, (7 May 2003); doi: 10.1117/12.488740

Published in SPIE Proceedings Vol. 5114:
Noise in Complex Systems and Stochastic Dynamics
Lutz Schimansky-Geier; Derek Abbott; Alexander Neiman; Christian Van den Broeck, Editor(s)