HIDDEN COVERS (WINGS) IN THE FRACTALS OF CHAOTIC SYSTEMS USING ADVANCED JULIA FUNCTION

IF 3.3 3区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Fractals-Complex Geometry Patterns and Scaling in Nature and Society Pub Date : 2023-10-14 DOI:10.1142/s0218348x23501256

MUHAMMAD MARWAN, MAOAN HAN, MAWIA OSMAN

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引用次数: 0

Abstract

In this work, we have adopted an advanced Julia-based function that helps not only in sorting out hidden wings in the fractals of chaotic systems, but can also generate an extra wing in chaotic systems based on a single wing. For verification, two examples Lorenz and modified stretch-twist-fold (STF) systems based on more than one wing, whereas chemical reaction-based chaotic system with a unique wing is considered. The existence of another wing in chaotic systems based on a single wing was a big question mark on the creation of multi-wings in the theory of fractals, but with the aid of advanced Julia functions, we have elaborated in detail that the existence of a second wing in such systems is also possible. The stretching and squeezing in the trajectories of fractals are also integral parts of our findings. Moreover, our study has solved another problem related to fractals. In the past, authors have shown multi-wings in chaotic systems with empty space inside all the time. In this study, we have shown for the first time that these inner spaces have special meaning due to the existence of hidden wings. Furthermore, we have shown that fractals can be divided into outer and inner wings, where the inner wings reflect the answer to the question about the empty space in between the outer wings of chaotic systems. For convenience, an extra file named as MiddleSpace.pdf is attached as a supplementary file to better understand the concept of covering empty space inside fractals.

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利用先进的Julia函数在混沌系统分形中隐藏罩(翼)

在这项工作中，我们采用了一种先进的基于julia的函数，它不仅可以在混沌系统的分形中整理出隐藏的翅膀，而且可以在单个翅膀的基础上在混沌系统中生成额外的翅膀。为了验证这一点，考虑了两个基于多个机翼的洛伦兹和改进的拉伸-扭转-折叠(STF)系统的例子，以及基于化学反应的具有唯一机翼的混沌系统。在基于单翼的混沌系统中，另一个翼的存在是分形理论中创建多翼的一个大问号，但借助先进的Julia函数，我们详细阐述了在这种系统中存在第二个翼也是可能的。分形轨迹中的拉伸和挤压也是我们研究结果的组成部分。此外，我们的研究还解决了与分形有关的另一个问题。过去，作者一直在内部空间为空的混沌系统中展示了多翼。在这项研究中，我们首次证明了这些内部空间由于隐藏翅膀的存在而具有特殊的意义。此外，我们还证明了分形可以分为外翼和内翼，其中内翼反映了混沌系统外翼之间空白空间的问题的答案。为了方便起见，附加了一个名为MiddleSpace.pdf的额外文件作为补充文件，以便更好地理解在分形中覆盖空白空间的概念。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Fractals-Complex Geometry Patterns and Scaling in Nature and Society 数学-数学跨学科应用

CiteScore

7.40

自引率

23.40%

发文量

319

审稿时长

>12 weeks

期刊介绍： The investigation of phenomena involving complex geometry, patterns and scaling has gone through a spectacular development and applications in the past decades. For this relatively short time, geometrical and/or temporal scaling have been shown to represent the common aspects of many processes occurring in an unusually diverse range of fields including physics, mathematics, biology, chemistry, economics, engineering and technology, and human behavior. As a rule, the complex nature of a phenomenon is manifested in the underlying intricate geometry which in most of the cases can be described in terms of objects with non-integer (fractal) dimension. In other cases, the distribution of events in time or various other quantities show specific scaling behavior, thus providing a better understanding of the relevant factors determining the given processes. Using fractal geometry and scaling as a language in the related theoretical, numerical and experimental investigations, it has been possible to get a deeper insight into previously intractable problems. Among many others, a better understanding of growth phenomena, turbulence, iterative functions, colloidal aggregation, biological pattern formation, stock markets and inhomogeneous materials has emerged through the application of such concepts as scale invariance, self-affinity and multifractality. The main challenge of the journal devoted exclusively to the above kinds of phenomena lies in its interdisciplinary nature; it is our commitment to bring together the most recent developments in these fields so that a fruitful interaction of various approaches and scientific views on complex spatial and temporal behaviors in both nature and society could take place.