The Proto-Lorenz System in Its Chaotic Fractional and Fractal Structure
Emile Franc Doungmo Goufo
Abstract
It is not common in applied sciences to realize simulations which depict fractal representation in attractors’ dynamics, the reason being a combination of many factors including the nature of the phenomenon that is described and the type of differential operator used in the system. In this work, we use the fractal-fractional derivative with a fractional order to analyze the modified proto-Lorenz system that is usually characterized by chaotic attractors with many scrolls. The fractal-fractional operator used in this paper is a combination of fractal process and fractional differentiation, which is a relatively new concept with most of the properties and features still to be known. We start by summarizing the basic notions related to the fractal-fractional operator. After that, we enumerate the main points related to the establishment of proto-Lorenz system’s equations, leading to the [Formula: see text]th cover of the proto-Lorenz system that contains [Formula: see text] scrolls ([Formula: see text]). The triple and quadric cover of the resulting fractal and fractional proto-Lorenz system are solved using the Haar wavelet methods and numerical simulations are performed. Due to the impact of the fractal-fractional operator, the system is able to maintain its chaotic state of attractor with many scrolls. Additionally, such attractor can self-replicate in a fractal process as the derivative order changes. This result reveals another great feature of the fractal-fractional derivative with fractional order.