The Effect of Fractional Order on Oscillatory Behavior of Scalar Fractional Delay Systems of Neutral Type

The undamped oscillations of a scalar fractional neutral system are studied in this paper. For this purpose, it is proved that this kind of systems can have the oscillatory behavior. The necessary and sufficient condition is proposed to determine that there exists a delay value for which the system oscillates. The frequency, amplitude and required delay for the undamped oscillations depend on the fractional order. However, the behavior of dependencies is not specified. Indeed, when the system has oscillatory behavior, it is bounded-input bounded-output stable for delay less than the required delay. Moreover, when the order is approaching to an integer value, the necessary and sufficient condition is proposed to determine that the system can oscillate with the bounded value of delay. The obtained results are verified with respect to the classical theorems. Finally, two numerical examples demonstrate the proposed conditions.