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Fractal Fract
Special Issues
Fractal Dimensions with Applications in the Real World
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Fractal Dimensions with Applications in the Real World

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Geometry".

Deadline for manuscript submissions: closed (30 June 2023) | Viewed by 1833

Special Issue Editors

Dr. Jun Luo

E-Mail Website
Guest Editor
College of Mathematics and Statistics, Chongqing University, Chongqing, China
Interests: fractal geometry; topology on fractals; Lipschitz equivalence; Gromov hyperbolic graphs
Dr. Xiaohao Cai

E-Mail Website
Guest Editor
School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK
Interests: computer vision; image processing; machine/deep learning; scientific computing
Special Issues, Collections and Topics in MDPI journals
Dr. Liang-Yi Huang

E-Mail Website
Guest Editor
College of Mathematics and Statistics, Chongqing University, Chongqing, China
Interests: multifractal analysis; Lipschitz equivalence; topology

Special Issue Information

Dear Colleagues,

The concept of fractals was introduced by B. Mandelbrot in the last 1970s as a class of highly irregular sets often presenting with infinite complexity, self-similarity and the nonintegral Hausdorff dimension. It has had a great impact in the development of mathematics and many other disciplines of science. In mathematics, fractal originates in chaos and dynamic systems, and soon after it was found that fractals appear in almost every area and are susceptible to systematic studies using classical and contemporary methods. In the last four decades, a large part of fractal research has been related to the dimension theories and structures of self-similar sets and measures.

The aim of this Special Issue is to present the up-to-date progress in fractal dimensions and their various applications to the real world. Topics invited for submission include, but are not limited to, the following:

  • Fractal dimensions of iterated function systems;
  • Fractal dimensions of self-similar measures;
  • Hausdorff dimension of fractal graphs;
  • Lipschitz equivalence of fractal sets;
  • Topological structures of fractal sets;
  • Diophantine approximation;
  • Fractal dimensions in dynamical systems;
  • Applications of fractal dimensions.

Dr. Jun Luo
Dr. Xiaohao Cai
Dr. Liang-Yi Huang
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • iterated function systems
  • self-similar sets
  • self-affine sets
  • fractal measures
  • fractal graphs
  • beta-expansions
  • Hausdorff dimension
  • box-counting dimension
  • Lipschitz equivalence
  • topological structures

Published Papers (2 papers)

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Research

19 pages, 10859 KiB  
Article
Reduced Order Modeling of System by Dynamic Modal Decom-Position with Fractal Dimension Feature Embedding
by Mingming Zhang, Simeng Bai, Aiguo Xia, Wei Tuo and Yongzhao Lv
Fractal Fract. 2024, 8(6), 331; https://doi.org/10.3390/fractalfract8060331 - 31 May 2024
Viewed by 300
Abstract
The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based [...] Read more.
The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based on fractal dimension (FD). The existing model selection methods lack interpretability and exhibit arbitrariness in choosing mode dimension truncation levels. To address these issues, this paper analyzes the geometric features of modes for the dimensional characteristics of dynamical systems. By calculating the box counting dimension (BCD) of modes and the correlation dimension (CD) and embedding dimension (ED) of the original dynamical system, it achieves guidance on the importance ranking of modes and the truncation order of modes in DMD. To validate the practicality of this method, it is applied to the reduction applications on the reconstruction of the velocity field of cylinder wake flow and the force field of compressor blades. Theoretical results demonstrate that the proposed selection technique can effectively characterize the primary dynamic features of the original dynamical systems. By employing a loss function to measure the accuracy of the reconstruction models, the computed results show that the overall errors of the reconstruction models are below 5%. These results indicate that this method, based on fractal theory, ensures the model’s accuracy and significantly reduces the complexity of subsequent computations, exhibiting strong interpretability and practicality. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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18 pages, 2306 KiB  
Article
New Properties and Sets Derived from the 2-Ball Fractal Dust
by Mario A. Aguirre-López, José Ulises Márquez-Urbina and Filiberto Hueyotl-Zahuantitla
Fractal Fract. 2023, 7(8), 612; https://doi.org/10.3390/fractalfract7080612 - 8 Aug 2023
Viewed by 784
Abstract
Due to their practicality and convenient parametrization, fractals derived from iterated function systems (IFSs) constitute powerful tools widely used to model natural and synthetic shapes. An IFS can generate sets other than fractals, extending its application field. Some of such sets arise from [...] Read more.
Due to their practicality and convenient parametrization, fractals derived from iterated function systems (IFSs) constitute powerful tools widely used to model natural and synthetic shapes. An IFS can generate sets other than fractals, extending its application field. Some of such sets arise from IFS fractals by adding minimal modifications to their defining rule. In this work, we propose two modifications to a fractal recently introduced by the authors: the so-called 2-ball fractal dust, which consists of a set of balls diminishing in size along an iterative process and delimited by an enclosing square. The proposed modifications are (a) adding a resizer parameter to introduce an interaction between the generator and generated ball elements and (b) a new fractal embedded into the 2-ball fractal dust, having the characteristic of filling zones not covered by the previous one. We study some numerical properties of both modified resulting sets to gain insights into their general properties. The resulting sets are geometrical forms with potential applications. Notably, the first modification generates an algorithm capable of producing geometric structures similar to those in mandalas and succulent plants; the second modification produces shapes similar to those found in nature, such as bubbles, sponges, and soil. Then, although a direct application of our findings is beyond the scope of this research, we discuss some clues of possible uses and extensions among which we can remark two connections: the first one between the parametrization we propose and the mandala patterns, and the second one between the embedded fractal and the grain size distribution of rocks, which is useful in percolation modeling. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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