Many engineering applications employ nonlinear systems, representable as a feedback interconnection of a linear time-invariant dynamic block and a periodic nonlinearity. Such models naturally describe phase-locked loops (PLLs), which are widely used for synchronization of built-in computer clocks, demodulation and frequency synthesis. Other example include, but are not limited to, dynamics of pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities, often referred to as synchronization systems, are usually featured by the existence of an infinite sequence of equilibria (stable or unstable). The central problem, concerning dynamics of synchronization systems, is the convergence of solutions to equilibria, treated in engineering applications as phase locking. In general, not any solution is convergent ('phase-locked'). This raises a natural question which oscillatory trajectories (such as e.g. periodic solutions) are possible. Even when the solution converges, the transient process can be unsatisfactory due to cycle slippings, leading to demodulation errors. In this paper, we address the mentioned three problems and offer novel criteria for phase locking, estimates for the number of slipped cycles and possible frequencies of periodic oscillations. The methods used in this paper are based on the method of integral quadratic constraints, stemming from Popov's technique of 'a priori integral indices.'