On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization

Abstract

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.

Mandelbrot set and Julia sets of fractional order

“…The difference revealed by Proposition 4 can be viewed in Figs. 4, where the IOM set and F OM set, for q = 1, are presented.…”

“…The fractional-order (FO) Mandelbrot and Julia sets in the sense of q-th Caputolike discrete fractional differences, for q ∈ (0, 1), generated by the quadratic complex (Mandelbrot) map f c (z) = z 2 + c, with z and c complex and starting from the initial value z 0 = 0 (the critical point), are introduced in [2] (see also [37,4,5,6]). The algorithms for generating FO sets, base on the known Mandelbrot set and Julia sets of Integer Order (IO) which, after they were discovered, still represent a huge source of inspiration for computer graphics programmers but also for mathematicians.…”

Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order

“…The fractional-order (FO) Mandelbrot and Julia sets in the sense of q-th Caputo-like discrete fractional differences, for q ∈ (0, 1), are fractal mathematical objects generated by the quadratic complex (Mandelbrot) map f c (z) = z 2 + c, with the z and c complexes, and starting from the initial value z 0 = 0 (the critical point), and are introduced in [1] (see also [2][3][4][5]). The algorithms for generating FO sets are based on the known Mandelbrot set and Julia sets of integer order (IO), which, after they were discovered, still represent a huge source of inspiration for computer graphics programmers as well as for mathematicians.…”