On Discrete Fractional Complex Gaussian Map: Fractal Analysis, Julia Sets Control, and Encryption Application

This work is devoted to present a generalized complex discrete fractional Gaussian map. Analytical and numerical analyses of the proposed map are conducted. The dynamical behaviors and stability of fixed points of the map are explored. The existence of fractal Mandelbrot and Julia sets is examined along with the corresponding fractal characteristics. The influences of the key parameters of the map and fractional order are examined. Moreover, nonlinear controllers are designed in the complex domain to control Julia sets generated by the map or to achieve synchronization between two Julia sets in master/slave configuration. Numerical simulations are provided to attain a deep understanding of nonlinear behaviors of the proposed map. Then, a suggested efficient chaos-based encryption technique is introduced by integrating the complicated dynamical behavior and fractal sets of the proposed map with the pseudo-chaos generated from the modified lemniscate hyperchaotic map.