The paper is devoted to asymptotic behavior of synchronization systems, i.e. Lur'e–type systems with periodic nonlinearities and infinite sets of equilibrum. This class of systems can not be efficiently investigated by standard Lyapunov functions. That is why for synchronization systems several new methods have been elaborated in the framework of Lyapunov direct method. Two of them: the method of periodic Lyapunov functions and the nonlocal reduction method, proved to be rather efficient. In this paper we combine these two methods and the Kalman-Yakubovich-Popov lemma to obtain new frequency–algebraic criteria ensuring Lagrange stability and the convergence of solutions.