On the Stability Domain of a Class of Linear Systems of Fractional Order

In this paper, the shape of the stability domain Sq for a class of difference systems defined by the Caputo forward difference operator delta(q) of order q is an element of (0,1) is numerically analyzed. It is shown numerically that due to of power of the negative base in the expression of the stability domain, in addition to the known cardioid-like shapes, Sq could present supplementary regions where the stability is not verified. The Mandelbrot map of fractional order is considered as an illustrative example. In addition, it is conjectured that for q < 0.5, the shape of Sq does not cover the main body of the underlying Mandelbrot set of fractional order as in the case of integer order.