Transient processes in synchronization systems governed by singularly perturbed Volterra equations

Many natural phenomena and engineering applications are based on synchronization between several periodic processes. A commonly known example is a phase-locked loop (PLL), that is, a feedback circuit providing synchronization between the endogenous oscillator and exogenous periodic signal in phase. Imprecise recovery of the signal´s phase leads to decoding and demodulation errors, so the properly designed PLL must provide the fast “phase-locking”, i.e. convergence of the phase shift (or error) to a steady value. In the simplest situation the error converges to the nearest equilibrium; in general, due to the initial conditions or disturbances, it can leave this basin of attraction and converge to another equilibrium. During this undesirable transient process, referred to as the cycle slipping, the phase shift is increased by a multiple of the period. The mechanical analog of a slipping PLL is a pendulum, making several turns around the point of suspension before stabilization at the lower equilibrium. In this paper we estimate the number of slipped cycles for a class of synchronization systems, governed by integro-differential Volterra equation with a scalar periodic nonlinearity and including, particularly, PLLs with discrete or distributed delays in the loop. These estimates are extended to singularly perturbed Volterra equations, having a small parameter at the higher derivative. Such models naturally arise if the system has “slow” and “fast” dynamics.