Abstract
Fractal structures are everywhere around us as they occur naturally or are artificially simulated. Applied mostly in engineering domains that include neural networks, fractal processes help fostering architectural design. For instance, fractal models are commonly used to design new machine learning algorithms for neural networks. Differential operators that can artificially trigger such fractal processes become a valuable asset for engineers. We use in this paper the fractal derivative combined to the fractional dynamic to analyze the chaotic proto-Lü system. That combined operator, known as the fractal-fractional derivative (FFD), is relatively new in the literature and has many features still to be discovered. The piecewise model of the proto-Lü system combining the fractal-fractional and classical derivatives is analyzed and solved numerically. In the study, we start by providing a succinct summary of fundamentals behind the FFD and equations of the proto-Lü system. The latter comprise different models with [Formula: see text] scrolls each ([Formula: see text]) corresponding to its nth cover. We focus on applying the FFD on the third cover of the proto-Lü system that is solved numerically via the Haar wavelets. Numerical simulations show the maintenance of the multiscroll chaotic attractors. The representations for the piecewise model also show the maintenance of the triple cover both in a stretched form and a self-similarity process. Additionally, we observe the capacity of those attractors to perform self-replication in a fractal structure that varies with the fractional-order derivative.