Relaxation cycles in systems with delay

Various examples of differential-difference equations encountered in applications show a rich variety of relaxation properties which, being important exceptions, do not permit the construction of a general theory. The authors study and give a new interpretation of a relaxation system which is typical from many points of view and which has been well known in the theory of quantum oscillators for 60 years. Two theorems on the existence of relaxation cycles are proved, and then a study of their stability properties is prepared, as usual, by constructing the asymptotics of the relaxation oscillations.