The phenomenon of the second order steady modes in the relay control systems with time delay

The second order relay control systems with delay (SOR-CSD) are considered. The notations of the frequency, oscillatory solutions and the second order steady modes for SORCSD are introduced. It is shown that the frequency of any oscillatory solution of SORCSD after a finite time interval becomes a constant. The sufficient conditions are found under which there exists a solution with any non-negative integer constant frequency which is called the steady mode. This means that any oscillatory solution coincides with some steady mode. It is shown that for any nonnegative frequency there exist the periodic steady modes of the autonomous SORCSD. It is justified that under some conditions any periodic solutions of the SORCSD with zero limit frequency are orbitally asymptotically stable.