Systems with periodic nonlinearities, referred to as pendulum–like systems or systems with cylindric phase space, naturally arise in many applications. Considered in the Euclidean space, such systems are usually featured by an infinite sequence of equilibria, none of them being globally stable. Hence the system’s “stability”, understood as convergence of every solution to one of the equilibria points (gradient-like behavior, or phase locking), cannot be examined by standard tools of nonlinear control, ensuring global asymptotic stability of a single equilibrium. Nevertheless, it appears that a modification of absolute stability methods, originating from the works of V.M. Popov, allows to establish efficient criteria for gradient-like behavior of pendulum-like system, which also imply the system’s robustness against a broad class of disturbances.