Upper and Lower Bounds for the Maximum Number of Frequencies That Can Be Generated by a Class of Fractional Oscillators

This paper deals with investigating the maximum number of frequencies generated by an oscillator in the case that a half number of integrators in the conventional dynamic structure of this oscillator are replaced by fractional integrators with an identical order. First, an upper bound for the maximum number of frequencies, which can exist in a steady state response of the oscillator, with respect to the number of integrators, is obtained. Then, on the basis of the recent advances in the field of determinant-based representations of bivariate polynomials, a lower bound is found for the maximum number of sinusoidal components in the steady state oscillations generated by the oscillator. A systematic algorithm is also introduced to find a dynamic structure and its dynamic matrices for realization of an oscillator, such that the number of distinct frequencies in its steady state oscillations is equal to the obtained lower bound. Numerical simulation results are presented to validate the introduced algorithm. Moreover, an idea for improving the lower bound is proposed. Finally, the paper is closed by describing some related open problems that invite future research works.