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.

FACTOR ANALYSIS OF GEOGRAPHIC COORDINATES AT POINTS OF THE CHANNEL OF A SMALL RIVER ON SPACE IMAGES

The purpose of the article is the analysis of asymmetric wavelets in binary relations between three coordinates at 290 characteristic points from the source to the mouth of the small river Irovka. The hypsometric characteristic is the most important property of the relief. The Irovka River belongs to a low level, at the mouth it is 89 m high, and at the source it is 148 m above sea level. Modeling of binary relations with latitude, longitude, and height has shown that local latitude receives the greatest quantum certainty. In this case, all paired regularities received a correlation coefficient of more than 0.95. Such a high adequacy of wave patterns shows that geomorphology can go over to the wave multiple fractal representation of the relief. The Irovka River is characterized by a small anthropogenic impact, therefore, the relief over a length of 69 km has the natural character of the oscillatory adaptation of a small river to the surface of the Vyatka Uval from its eastern side. This allows us to proceed to the analysis of the four tributaries of the small river Irovka, as well as to model the relief of the entire catchment basin of 917 km2. The greatest adequacy with a correlation coefficient of 0.9976 was obtained by the influence of latitude on longitude, that is, the geographical location of the relief of the river channel with respect to the geomorphology of the Vyatka Uval. In second place with a correlation of 0.9967 was the influence of the height of the points of the channel of the small river on local longitude and it is also mainly determined by the relief of the Vyatka Uval. In third place was the effect of latitude on height with a correlation coefficient of 0.9859. And in last sixth place is the inverse effect of altitude on local latitude in the North-South direction.

Fractal representation of the power demand based on topological properties of julia sets

In a power system, the load demand considers two components such as the real power (P) because of resistive elements, and the reactive power (Q) because inductive or capacitive elements. This paper presents a graphical representation of the electric power demand based on the topological properties of the Julia Sets, with the purpose of observing the different graphic patterns and relationship with the hourly load consumptions. An algorithm that iterates complex numbers related to power is used to represent each fractal diagram of the load demand. The results show some representative patterns related to each value of the power consumption and similar behaviour in the fractal diagrams, which allows to understand consumption behaviours from the different hours of the day. This study allows to make a relation among the different consumptions of the day to create relationships that lead to the prediction of different behaviour patterns of the curves.

Multimodality of the category EMOTIONS in art discourse: a fractal semiotics perspective

This article focuses on revealing the ontological peculiarities of the multimodal nature of the verbally represented category EMOTIONS in the ecology of art discourse. On the grounds of philosophy of emotion, psychology of art, cognitive theories of emotion and B. Mandelbrot’s theory of fractals to complement the linguocognitive, cognitive-discursive and semiotics approaches, we elaborate an integrative evidence framework of the fractality of the category EMOTIONS in the circumflex of its conceptual components JOY, SADNESS, INTEREST, SURPRISE, ANGER, FEAR, DISGUST as natural fractals, actualized in modern art discourse in matrices of their conceptual features EXPERIENCE STIMULUS, APPRAISAL, EMERGENCE TIME, DIRECTION. A special emphasis has been laid on both the essence of art discourse, implying discourses on art and art as discourse, being an environment of conceptualizing and verbalizing aesthetic emotions, and on the role of the cognitive mechanisms of metaphors and metonymies in multimodal objectivizations of the category EMOTIONS fractals in discourses of visual arts and music (installations and tone painting). The relation between emotion, cognition, semiotics, art and language has been specified. As a result of the analysis based on the core multimodal research concepts of modes, semiotic resource, modal affordance and intersemiotic relations we have found out that verbalized category EMOTIONS conceptual ingredients are anchored in art perception, acquiring their fractal representation in clusters of modalities. The latter comprise the alethic, deontic, axiological, temporal, spatial, epistemic ones, getting recursively realized verbally, visually, aurally, linked with synesthesia, including chromesthesia, ideasthesia, spatial sequence, triggered by conceptualized and verbalized feeling experience response. With this work we demonstrate the value of fractal semiotics approach as a heuristic device to study multimodal categories.