The Fractional Variable-Order Grassi–Miller Map: Chaos, Complexity, and Control

In the topic of discrete variable-order systems governed by fractional difference equations, this study makes a significant contribution by introducing two innovative variable-order versions of the fractional Grassi–Miller system. These new formulations are aimed at deepening our understanding of the complex dynamics that such systems exhibit. The research specifically delves into the chaotic dynamical behaviors manifested by these systems: one version being the fractional Grassi–Miller map with commensurate variable order and the other being the fractional Grassi–Miller map with incommensurate variable order. To provide a comprehensive analysis, this study incorporates a variety of variable orders, encompassing both exponential and sinusoidal functions. These variable orders are crucial in exploring how different functional forms influence the behavior of the system. By varying these orders, the research seeks to uncover the patterns and chaotic dynamics that emerge under different conditions. A suite of advanced numerical methods is employed to rigorously analyze and validate the presence of chaotic attractors in these newly proposed variable fractional versions of the Grassi–Miller system. The methods used include bifurcation diagrams, phase portraits, Lyapunov exponents, approximate entropy, C₀ complexity, and 0–1 test for chaos. Through the application of these numerical methods, the study thoroughly validates the existence of chaotic attractors in the proposed variable fractional versions of the Grassi–Miller system. The findings underscore the rich and complex behaviors that arise from different variable orders, offering new insights into the dynamics of fractional-order systems.