Fractal control of Julia sets generated by discrete fractional competitive model

This article explores the fractal behavior of a general non-linear discrete fractionalorder competitive model. In particular, we focus on a discrete fractional competitive model that illustrates the growth of two competing species influenced by toxic compounds. The analysis in this paper can be extended to other discrete fractional competitive models. To date, such models influenced by toxic compounds have not been studied from a fractal perspective, nor has the innovative control method been applied to the control of Julia sets in discrete models. Initially, we presented the discrete fractional competitive model utilizing the Gr¨unwald-Letnikov operator. Following that, we introduced the Julia set produced by a discrete fractional competitive model with the help of a modified escape-time algorithm suitable for the fractional order scenario. The fractal dynamic characteristics and the influence of memory parameters on the Julia set are illustrated through graphical exploration and fractal dimension analysis. Controlling Julia sets of the discrete fractional map is achieved by stabilizing the equilibrium point. The stability of these equilibrium points and consequently, the Julia sets are controlled using two distinct control methods. As the control parameter values diminish in the gradient control method, the Julia set related to the controlled system begins to shrink. This effect is a result of the instability of the equilibrium point. We developed a control scheme that helps in the control of the Julia set. When the control parameter values increase within a specific range, the Julia set in the second new state control technique compresses both vertically and horizontally. Again, this is attributed to the instability of the equilibrium point. To synchronize the Julia sets of two similar systems with different system parameters, a new synchronous controller featuring an arbitrary constant has been developed. The effectiveness of this synchronous controller is evaluated using numerical simulations. Our methodologies and various theoretical studies are solidified with the help of numerical simulations. This analysis will also apply to analogous types of mathematical models with different dimensions.