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This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings. Moreover, the stability of the zero fixed point and symmetry are examined by theoretical analysis, and it is proved that the map generates diverse nonlinear traits comprising multistability, chaos, and hyperchaos, which is confirmed by phase attractors in 2D and 3D space, Lyapunov exponents ( L E s ) analysis and bifurcation diagrams; also, 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. In addition, a nonlinear controller is introduced to stabilize the symmetry map and synchronize a duo of unified symmetry maps. Finally, numerical results are provided to illustrate the findings. [ABSTRACT FROM AUTHOR]

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C 0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map's states in commensurate and incommensurate cases. [ABSTRACT FROM AUTHOR]