Characteristic Roots of a Class of Fractional Oscillators

The fundamental theorem of algebra determines the number of characteristic roots of an ordinary differential equation of integer order. This may cease to be true for a differential equation of fractional order. The results given in this paper suggest that the number of the characteristic roots of a class of oscillators of fractional order may in general be infinitely great. Further, we infer that it may also be the case for the characteristic roots of a differential equation of fractional order greater than 1. The relationship between the range of the fractional order and the locations of characteristic roots of oscillators in the complex plane is considered.