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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.

A discreet fractional-order Cournot–Bertrand competition duopoly game is introduced based on the fractional-order difference calculus of the Caputo operator. The model is designed when players can make long memory decisions. The local stability of equilibrium points is discussed for the proposed model. Some numerical simulations explore the model’s bifurcation and chaos by employing bifurcation diagrams, phase portraits, maximal Lyapunov exponents, and time series. According to our findings, the fractional-order parameter has an effect on the game’s stability and dynamics.

This paper attempts to further extend the results of dynamical analysis carried out on a recent 4D Lorenz-type hyperchaotic system while exploring new analytical results concerns its local and global dynamics. In particular, the equilibrium points of the system along with solution’s continuous dependence on initial conditions are examined. Then, a detailedZ2symmetrical Bogdanov-Takens bifurcation analysis of the hyperchaotic system is performed. Also, the possible first integrals and global invariant surfaces which exist in system’s phase space are analytically found. Theoretical results reveal the rich dynamics and the complexity of system behavior. Finally, numerical simulations and a proposed circuit implementation of the hyperchaotic system are provided to validate the present analytical study of the system.