Analysis of undamped oscillations generated by marginally stable fractional order systems

In this paper, undamped oscillations generated by fractional order systems are studied. At first, the trajectories of marginally stable fractional order systems with commensurate order are investigated and the differences with the integer order case are highlighted. Then, amplitude and frequency of oscillations in the steady state are analytically determined for these systems. It is shown that for a linear time invariant commensurate fractional order system, the maximum number of frequencies exists in the Fourier spectrum of the oscillations is half of its inner dimension, which is the same property as in the linear time invariant integer order systems. However, we show that this property is not necessarily exists for the incommensurate fractional order systems. Finally, the paper is closed by posing an open problem in this filed.