Systems with Periodic Nonlinearities

Closed quasilinear control systems with a scalar periodic nonlinearity are studied. Theorems on the existence of sequences of periodic oscillations are presented. For autonomous systems, conditions for the existence of sequences of cycles are given and it is assumed that the amplitudes of the cycles in this sequence increase to infinity, while the periods converge to a fixed value. Perturbations of the original autonomous system by a periodic input of the same period are studied. conditions for the existence of infinite unbounded families of cycles and forced oscillations are derived for equations with usual periodic nonlinearities. Infinite sequence of periodic solutions such that their amplitudes tend to infinity and their periods tend to 2π are found. It is proved that for natural f, the function tends at infinity to a finite limit, frequently, to zero.