The proto-Lorenz system in its chaotic fractional and fractal structure. (English) Zbl 1452.37086
Summary: 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 \(n\)th cover of the proto-Lorenz system that contains \(n\) scrolls \((n\in\mathbb N)\). 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.
MSC:
| 37M22 | Computational methods for attractors of dynamical systems |
| 37M05 | Simulation of dynamical systems |
| 37C45 | Dimension theory of smooth dynamical systems |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 26A33 | Fractional derivatives and integrals |
| 28A80 | Fractals |
Keywords:
Lorenz model; proto-Lorenz system; fractal-fractional model; multiscroll chaotic attractor; fractalsReferences:
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