Super-high-frequency oscillations in a discontinuous dynamic system with time delay

AbstractWe study oscillations in the discontinuous dynamic system with time delay $$\dot x(t) = - sign x(t - 1) + F(x(t),t), t \geqslant 0$$ . This is a typical model of relay feedback with delay. It is known that stable modes in this system have a bounded oscillation frequency. Here we consider transient processes and obtain the following result: under some restrictions ofF, the average oscillation frequency of any solution becomes finite after a period of time, i.e. super-high-frequency oscillations (with infinite frequency) exist only in a finite time interval. Moreover, we give an effective upper bound on the length of this interval.

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