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Using fractional difference equations to describe fractional and variable-order maps, this manuscript discusses the dynamics of the discrete 4D sinusoidal feedback sine iterative chaotic map with infinite collapse (ICMIC) modulation map (SF-SIMM) with fractional-order. Also, it presents a novel variable-order version of SF-SIMM and discusses their chaotic dynamic behavior by employing a distinct function for the variable fractional-order. To establish the existence of chaos in the suggested discrete SF-SIMM, some numerical methods such as phase plots, bifurcation and largest Lyapunov exponent diagrams,  C 0  complexity and 0–1 test are utilized. After that, two different control schemes are used for the conceived discrete system. The states are stabilized and asymptotically forced towards zero by the first controller. The second controller is used to synchronize a pair of maps with non–identical parameters. Finally, MATLAB simulations will be executed to confirm the results provided. [ABSTRACT FROM AUTHOR]

This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the interplay of different fractional orders in the proposed map. Furthermore, the study employs the sample entropy test to quantify complexity and validate the presence of chaos. Non-linear controllers are also presented to stabilize and synchronize the model. The research emphasizes the system's sensitivity to the fractional order parameters, leading to distinct dynamic patterns and stability regimes. The memristor-based chaotic map exhibits rich and intricate behavior, making it an interesting and important area of research. [ABSTRACT FROM AUTHOR]