The Proto-Lorenz System in Its Chaotic Fractional and Fractal Structure

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.

What does ‘Culture’ Mean in French? A Theoretical Mapping and Fractal Analysis of the Sociology of Culture in France

This article charts the development of the sociology of culture in France. First, it examines the hypothesis of a French model, putting into perspective the correlation between cultural policies and dedicated sociological inquiries at the end of the 1950s. ‘Culture’ is one of the oldest fields of research in France, and current research still derives from the same anthropological matrix. Yet French sociologists present themselves as part of a divided and competitive academic domain. This article, based on an encompassing review of the literature as well as on in-depth interviews, accordingly distinguishes eight different ‘schools’ – organized around pre-eminent academics, concept producers and resource providers – as well as circles of collaboration. Whilst these circles organize their theoretical activity around emblems (with the word ‘culture’ referring to different conceptual sets) the social relations in their midst are organized around dyads, which usually transition from positive collaboration to rivalry. The article highlights the importance of these divisions as a fractal process and as boundary work for scientific production. From this perspective, the sociology of culture in France could be described as a large and extensive system of concepts and collaborations developed within small groups, within and between which, as with all ‘cultural’ matters, symbolic activity is the key basis for social status.

A fractal process of hydrogen diffusion in a-Si:H with exponential energy distribution

Hydrogen diffusion in a-Si:H with exponential distribution of the states in energy exhibits the fractal structure. It is shown that a probability [Formula: see text] of the pausing time [Formula: see text] has a form of [Formula: see text] ([Formula: see text]: fractal dimension). It is shown that the fractal dimension [Formula: see text]/[Formula: see text] ([Formula: see text]: hydrogen temperature, [Formula: see text]: a temperature corresponding to the width of exponential distribution of the states in energy) is in agreement with the Hausdorff dimension. A fractal graph for the case of [Formula: see text] is like the Cantor set. A fractal graph for the case of [Formula: see text] is like the Koch curves. At [Formula: see text], hydrogen migration exhibits Brownian motion. Hydrogen diffusion in a-Si:H should be the fractal process.

He’s fractional derivative and its application for fractional Fornberg-Whitham equation

Fractional Fornberg-Whitham equation with He?s fractional derivative is studied in a fractal process. The fractional complex transform is adopted to convert the studied fractional equation into a differential equation, and He's homotopy perturbation method is used to solve the equation.