In this paper we consider control systems with periodic nonlinearities characterized by countable sets of equilibria, both Lyapunov stable and unstable. The simplest example of such a system is the mathematical pendulum; therefore, these systems are often called “pendulum-like” systems. In pendulum-like systems, the very concept of stability differs from that in systems with a unique equilibrium. Stability is defined as the convergence of any solution to a certain equilibrium. For stable pendulum-like systems, the problem of cycle slipping arises. In the case of a mathematical pendulum, the number of slipped cycles corresponds to the number of rotations of the pendulum around its suspension point. In general, it represents the distance between the initial value of the input and its limit value. In this paper, we obtain frequency-domain estimates for the number of slipped cycles in infinite-dimensional systems using the Popov method of a priori integral indices. These estimates are tighter than those established in previous works. The paper presents an expanded version of the talk delivered at the International Conference on Physics and Control (PhysCon 2024).