On the stability domain of a class of linear systems of fractional order

TL;DR

This paper numerically investigates the shape of the stability domain for a class of fractional order difference systems, revealing complex geometries and conjecturing differences from integer order cases.

Contribution

It provides a numerical analysis of the stability domain for fractional difference systems and explores the influence of the order q on its shape, including conjectures about its relation to the Mandelbrot set.

Findings

Stability domain S^q can have complex, non-cardioid shapes.

Additional regions of instability are identified beyond known shapes.

For q<0.5, the stability domain may not cover the main Mandelbrot set body.

Abstract

In this paper, the shape of the stability domain S^q for a class of difference systems defined by the Caputo forward difference operator D^q of order q\in (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, S^q 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 S^q does not cover the main body of the underlying Mandelbrot set of fractional order as in the case of integer order.