Gingerbread-man Discrete System Formulated by the k-Symbol Fractional Calculus
Rabha Waell Ibrahim · Apple Academic Press eBooks · 2025
In the present study, we extend the gingerbread-man map by utilizing the concept of a k -symbol Caputo derivative within the framework of fractional calculus. A two-dimensional chaotic map, dependent on the k -symbol gingerbread-man map and the fractional flow map (K-GM), is proposed. We investigate some of the important dynamics of these maps and analyze the conditions for the stability of variable fractional dynamic systems. As a result, we present the K-GM with stability solution specifications based on the stability of the system’s generating polynomial. Additionally, the Hermite matrix-based predecessor of the system considers the symmetry stability criterion. Furthermore, to achieve system stabilization, we propose incorporating these maps into control processes. In this study, both onedimensional (1D) and two-dimensional (2D) controller rules are assumed. 289 The symmetry and stability of the controller system alternatives are examined, and we also present asymmetric stability control systems. The linearization approach is employed to compute synchronization errors for the two- and four-dimensional controller laws. The primary findings of this investigation are demonstrated through mathematical models.