Fractional Order Curves

This paper continues the subject of symmetry breaking of fractional-order maps, previously addressed by one of the authors. Several known planar classes of curves of integer order are considered and transformed into their fractional order. Several known planar classes of curves of integer order are considered and transformed into their fractional order. For this purpose, the Grunwald–Letnikov numerical scheme is used. It is shown numerically that the aesthetic appeal of most of the considered curves of integer order is broken when the curves are transformed into fractional-order variants. The considered curves are defined by parametric representation, Cartesian representation, and iterated function systems. To facilitate the numerical implementation, most of the curves are considered under their affine function representation. In this way, the utilized iterative algorithm can be easily followed. Besides histograms, the entropy of a curve, a useful numerical tool to unveil the characteristics of the obtained fractional-order curves and to compare them with their integer-order counterparts, is used. A Matlab code is presented that can be easily modified to run for all considered curves.