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Fractal Fract
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Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering
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Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering

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

Deadline for manuscript submissions: 20 July 2025 | Viewed by 16112

Special Issue Editors

Dr. Alexandra M.S.F. Galhano

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Faculdade de Ciências Naturais, Engenharias e Tecnologias, Universidade Lusófona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal
Interests: systems modelling; dynamics; multidimensional scaling; fractional calculus
Prof. Dr. Sergio Adriani David

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Guest Editor
Faculdade de Zootecnia e Engenharia de Alimentos da USP, University of São Paulo, Av. Duque de Caxias-Norte, 225, Jardim Elite, Pirassununga 13635-900, SP, Brazil
Interests: fractional order systems; fractional behaviour; fractional modelling for time series; fractional modelling in econophysics; fractional modelling in biological systems; nonlinear phenomena; fractals and chaos
Special Issues, Collections and Topics in MDPI journals
Dr. António Lopes

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Guest Editor
Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
Interests: complex systems modelling; automation and robotics; fractional order systems modelling and control; data analysis and visualization; machine learning
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Many problems in classical and quantum physics, statistical physics, engineering, biology, psychology, economics, and finance are of a global nature (rather than simply local) and their behavior is often characterized by long-range correlations in the time–space domain, memory effects, fractality, and power law dynamics. The fractional calculus and fractional processes have been extensively adopted in various areas and have become one of the most useful approaches to deal with particular properties of non-locality and representation of (long) memory effects in a myriad of applied sciences. Indeed, the fractional paradigm applies not only to calculus but also to stochastic processes. Moreover, big data analysis, organization, retrieval, and modeling are important tools for a computational approach to address complex, fractal, and fractional dynamics.

This Special Issue (SI) is important, not only to present the state of the art for complex, fractal, and fractional dynamics and their applications, but also to reveal the potential and the extension of those tools to model real world phenomena. Original, rigorous, and high-quality contributions are welcome to this SI and should fit the scope of the journal. Potential authors should address topics that include, but are not limited to, the following:

  • Memory (univariate and multivariate) models;
  • Complex and fractional modeling for time series;
  • Complex and fractional modeling in econophysics;
  • Complex and fractional approaches in biosystems and biophysics;
  • Complex and fractional dynamics in oncology;
  • Mathematical psychology;
  • Fractals;
  • Fractal-Fractional order mathematical models;
  • Fractional non-linear dynamics and chaos;
  • Big data in complex and fractional dynamics;
  • Fractional order advanced control systems: cyber-physical systems, machine learning, robotics, mechanical systems, etc.

Dr. Alexandra M.S.F. Galhano
Prof. Dr. Sergio Adriani David
Dr. António Lopes
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

  • fractional-order systems
  • memory
  • econophysics
  • time series
  • finance
  • economics, big data
  • complex systems
  • biosystems
  • fractals

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Published Papers (10 papers)

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Research

17 pages, 2057 KiB  
Article
A Fractional Time–Space Stochastic Advection–Diffusion Equation for Modeling Atmospheric Moisture Transport at Ocean–Atmosphere Interfaces
by Behrouz Parsa Moghaddam, Mahmoud A. Zaky, António Mendes Lopes and Alexandra Galhano
Fractal Fract. 2025, 9(4), 211; https://doi.org/10.3390/fractalfract9040211 - 28 Mar 2025
Cited by 1 | Viewed by 429
Abstract
This study introduces a novel one-dimensional Fractional Time–Space Stochastic Advection–Diffusion Equation that revolutionizes the modeling of moisture transport within atmospheric boundary layers adjacent to oceanic surfaces. By synthesizing fractional calculus, advective transport mechanisms, and pink noise stochasticity, the proposed model captures the intricate [...] Read more.
This study introduces a novel one-dimensional Fractional Time–Space Stochastic Advection–Diffusion Equation that revolutionizes the modeling of moisture transport within atmospheric boundary layers adjacent to oceanic surfaces. By synthesizing fractional calculus, advective transport mechanisms, and pink noise stochasticity, the proposed model captures the intricate interplay between temporal memory effects, non-local turbulent diffusion, and the correlated-fluctuations characteristic of complex ocean–atmosphere interactions. The framework employs the Caputo fractional derivative to represent temporal persistence and the fractional Laplacian to model non-local turbulent diffusion, and incorporates a stochastic term with a 1/f power spectral density to simulate environmental variability. An efficient numerical solution methodology is derived utilizing complementary Fourier and Laplace transforms, which elegantly converts spatial fractional operators into algebraic expressions and yields closed-form solutions via Mittag–Leffler functions. This method’s application to a benchmark coastal domain demonstrates that stronger advection significantly increases the spatial extent of conditions exceeding fog formation thresholds, revealing advection’s critical role in moisture transport dynamics. Numerical simulations demonstrate the model’s capacity to reproduce both anomalous diffusion phenomena and realistic stochastic variability, while convergence analysis confirms the numerical scheme’s robustness against varying noise intensities. This integrated fractional stochastic framework substantially advances atmospheric moisture modeling capabilities, with direct applications to meteorological forecasting, coastal climate assessment, and environmental engineering. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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45 pages, 5094 KiB  
Article
New Class of Complex Models of Materials with Piezoelectric Properties with Differential Constitutive Relations of Fractional Order: An Overview
by Katica R. (Stevanović) Hedrih
Fractal Fract. 2025, 9(3), 170; https://doi.org/10.3390/fractalfract9030170 - 11 Mar 2025
Viewed by 483
Abstract
Rheological complex models of various elastoviscous and viscoelastic fractional-type substances with polarized piezoelectric properties are of interest due to the widespread use of viscoelastic–plastic bodies under loading. The word “overview” used in the title means and corresponds to the content of the manuscript [...] Read more.
Rheological complex models of various elastoviscous and viscoelastic fractional-type substances with polarized piezoelectric properties are of interest due to the widespread use of viscoelastic–plastic bodies under loading. The word “overview” used in the title means and corresponds to the content of the manuscript and aims to emphasize that it presents an overview of a new class of complex rheological models of the fractional type of ideal elastoviscous, as well as viscoelastic, materials with piezoelectric properties. Two new elementary rheological elements were introduced: a rheological basic Newton’s element of ideal fluid fractional type and a basic Faraday element of ideal elastic material with the property of polarization under mechanical loading and piezoelectric properties. By incorporating these newly introduced rheological elements into classical complex rheological models, a new class of complex rheological models of materials with piezoelectric properties described by differential fractional-order constitutive relations was obtained. A set of seven new complex rheological models of materials are presented with appropriate structural formulas. Differential constitutive relations of the fractional order, which contain differential operators of the fractional order, are composed. The seven new complex models describe the properties of ideal new materials, which can be elastoviscous solids or viscoelastic fluids. The purpose of the work is to make a theoretical contribution by introducing, designing, and presenting a new class of rheological complex models with appropriate differential constitutive relations of the fractional order. These theoretical results can be the basis for further scientific and applied research. It is especially important to point out the possibility that these models containing a Faraday element can be used to collect electrical energy for various purposes. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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35 pages, 2120 KiB  
Article
Fractional Transfer Entropy Networks: Short- and Long-Memory Perspectives on Global Stock Market Interactions
by Ömer Akgüller, Mehmet Ali Balcı, Larissa Margareta Batrancea and Lucian Gaban
Fractal Fract. 2025, 9(2), 69; https://doi.org/10.3390/fractalfract9020069 - 23 Jan 2025
Cited by 1 | Viewed by 874
Abstract
This study addresses the challenge of capturing both short-run volatility and long-run dependencies in global stock markets by introducing fractional transfer entropy (FTE), a new framework that embeds fractional calculus into transfer entropy. FTE allows analysts to tune memory parameters and thus observe [...] Read more.
This study addresses the challenge of capturing both short-run volatility and long-run dependencies in global stock markets by introducing fractional transfer entropy (FTE), a new framework that embeds fractional calculus into transfer entropy. FTE allows analysts to tune memory parameters and thus observe how different temporal emphases reshape the network of directional information flows among major financial indices. Empirical evidence reveals that when short-memory effects dominate, markets swiftly incorporate recent news, creating networks that adapt quickly but remain vulnerable to transient shocks. In contrast, balanced memory parameters yield a more stable equilibrium, blending immediate reactions with persistent structural ties. Under long-memory configurations, historically entrenched relationships prevail, enabling established market leaders to remain central despite ongoing fluctuations. These findings demonstrate that FTE uncovers nuanced dynamics overlooked by methods focusing solely on either current events or deep-rooted patterns. Although the method relies on price returns and does not differentiate specific shock types, it offers a versatile tool for investors, policymakers, and researchers to gauge financial stability, evaluate contagion risk, and better understand how ephemeral signals and historical legacies jointly govern global market connectivity. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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37 pages, 9612 KiB  
Article
Rheological Burgers–Faraday Models and Rheological Dynamical Systems with Fractional Derivatives and Their Application in Biomechanics
by Katica R. (Stevanović) Hedrih and Andjelka N. Hedrih
Fractal Fract. 2024, 8(12), 742; https://doi.org/10.3390/fractalfract8120742 - 16 Dec 2024
Cited by 1 | Viewed by 943
Abstract
Two rheological Burgers–Faraday models and rheological dynamical systems were created by using two new rheological models: Kelvin–Voigt–Faraday fractional-type model and Maxwell–Faraday fractional-type model. The Burgers–Faraday models described in the paper are new models that examine the dynamical behavior of materials with coupled fields: [...] Read more.
Two rheological Burgers–Faraday models and rheological dynamical systems were created by using two new rheological models: Kelvin–Voigt–Faraday fractional-type model and Maxwell–Faraday fractional-type model. The Burgers–Faraday models described in the paper are new models that examine the dynamical behavior of materials with coupled fields: mechanical stress and strain and the electric field of polarization through the Faraday element. The analysis of the constitutive relation of the fractional order for Burgers–Faraday models is given. Two Burgers–Faraday fractional-type dynamical systems were created under certain approximations. Both rheological Burgers-Faraday dynamic systems have two internal degrees of freedom, which are introduced into the system by each standard light Burgers-Faraday bonding element. It is shown that the sequence of bonding elements in the structure of the standard light Burgers-Faraday bonding element changes the dynamic properties of the rheological dynamic system, so that in one case the system behaves as a fractional-type oscillator, while in the other case, it exhibits a creeping or pulsating behavior under the influence of an external periodic force. These models of rheological dynamic systems can be used to model new natural and synthetic biomaterials that possess both viscoelastic/viscoplastic and piezoelectric properties and have dynamical properties of stress relaxation. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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18 pages, 1149 KiB  
Article
Approaching Multifractal Complexity in Decentralized Cryptocurrency Trading
by Marcin Wątorek, Marcin Królczyk, Jarosław Kwapień, Tomasz Stanisz and Stanisław Drożdż
Fractal Fract. 2024, 8(11), 652; https://doi.org/10.3390/fractalfract8110652 - 11 Nov 2024
Cited by 3 | Viewed by 2088
Abstract
Multifractality is a concept that helps compactly grasp the most essential features of financial dynamics. In its fully developed form, this concept applies to essentially all mature financial markets and even to more liquid cryptocurrencies traded on centralized exchanges. A new element that [...] Read more.
Multifractality is a concept that helps compactly grasp the most essential features of financial dynamics. In its fully developed form, this concept applies to essentially all mature financial markets and even to more liquid cryptocurrencies traded on centralized exchanges. A new element that adds complexity to cryptocurrency markets is the possibility of decentralized trading. Based on the extracted tick-by-tick transaction data from the Universal Router contract of the Uniswap decentralized exchange, from 6 June 2023 to 30 June 2024, the present study using multifractal detrended fluctuation analysis (MFDFA) shows that even though liquidity on these new exchanges is still much lower compared to centralized exchanges, convincing traces of multifractality are already emerging in this new trading as well. The resulting multifractal spectra are, however, strongly left-side asymmetric, which indicates that this multifractality comes primarily from large fluctuations, and small ones are more of the uncorrelated noise type. What is particularly interesting here is the fact that multifractality is more developed for time series representing transaction volumes than rates of return. On the level of these larger events, a trace of multifractal cross-correlations between the two characteristics is also observed. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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19 pages, 1159 KiB  
Article
Formation of Optical Fractals by Chaotic Solitons in Coupled Nonlinear Helmholtz Equations
by M. Mossa Al-Sawalha, Saima Noor, Mohammad Alqudah, Musaad S. Aldhabani and Rasool Shah
Fractal Fract. 2024, 8(10), 594; https://doi.org/10.3390/fractalfract8100594 - 10 Oct 2024
Viewed by 1025
Abstract
In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CNHEs). This system models the transmission of optical solitons and [...] Read more.
In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CNHEs). This system models the transmission of optical solitons and coupled wave packets in nonlinear optical fibers and describes transverse effects in nonlinear fiber optics. Initially, a complex transformation is used to convert the model into a single Nonlinear Ordinary Differential Equation (NODE), from which hyperbolic, exponential, rational, trigonometric, and rational hyperbolic solutions are produced. In order to better understand the physical dynamics, we offer several 3D, contour, and 2D illustrations for the independent selections of physical parameter values. These illustrations highlight the graphic behaviour of some optical solitons and demonstrate that, under certain constraint conditions, acquired optical solitons lose their stability when they approach an axis and display periodic-axial perturbations, which lead to the generation of optical fractals. As a framework, the generated optical solitons have several useful applications in the field of telecommunications. Furthermore, our suggested RMESEM demonstrates its use by broadening the spectrum of optical soliton solutions, offering important insights into the dynamics of the CNHEs, and suggesting possible applications in the management of nonlinear models. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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17 pages, 356 KiB  
Article
Information Properties of Consecutive Systems Using Fractional Generalized Cumulative Residual Entropy
by Mohamed Kayid and Mansour Shrahili
Fractal Fract. 2024, 8(10), 568; https://doi.org/10.3390/fractalfract8100568 - 28 Sep 2024
Cited by 2 | Viewed by 685
Abstract
We investigate some information properties of consecutive k-out-of-n:G systems in light of fractional generalized cumulative residual entropy. We firstly derive a formula to compute fractional generalized cumulative residual entropy related to the system’s lifetime and explore its preservation properties in [...] Read more.
We investigate some information properties of consecutive k-out-of-n:G systems in light of fractional generalized cumulative residual entropy. We firstly derive a formula to compute fractional generalized cumulative residual entropy related to the system’s lifetime and explore its preservation properties in terms of established stochastic orders. Additionally, we obtain useful bounds. To aid practical applications, we propose two nonparametric estimators for the fractional generalized cumulative residual entropy in these systems. The efficiency and performance of these estimators are illustrated using simulated and real datasets. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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12 pages, 1090 KiB  
Article
Dynamics of the Traveling Wave Solutions of Fractional Date–Jimbo–Kashiwara–Miwa Equation via Riccati–Bernoulli Sub-ODE Method through Bäcklund Transformation
by M. Mossa Al-Sawalha, Saima Noor, Mohammad Alqudah, Musaad S. Aldhabani and Roman Ullah
Fractal Fract. 2024, 8(9), 497; https://doi.org/10.3390/fractalfract8090497 - 23 Aug 2024
Viewed by 987
Abstract
The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement [...] Read more.
The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement of the characterization of dynamic waves while providing better modeling ability compared to integer types of derivatives. The solutions of the above-mentioned time–space fractional Date–Jimbo–Kashiwara–Miwa equation have tremendous importance in numerous scientific scenarios. The regular dynamical wave solutions of the aforementioned equation encompass three fundamental functions: trigonometric, hyperbolic, and rational functions will be among the topics covered. These solutions are graphically classified into three categories: compacton kink solitary wave solutions, kink soliton wave solutions and anti-kink soliton wave solutions. In addition, to explore the impact of the fractional parameter (α) on those solutions, 2D plots are utilized, while 3D plots are applied to present the solutions involving the integer-order derivatives. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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23 pages, 2057 KiB  
Article
Navigating Choppy Waters: Interplay between Financial Stress and Commodity Market Indices
by Haji Ahmed, Faheem Aslam and Paulo Ferreira
Fractal Fract. 2024, 8(2), 96; https://doi.org/10.3390/fractalfract8020096 - 4 Feb 2024
Cited by 3 | Viewed by 2603
Abstract
Financial stress can have significant implications for individuals, businesses, asset prices and the economy as a whole. This study examines the nonlinear structure and dynamic changes in the multifractal behavior of cross-correlation between the financial stress index (FSI) and four well-known commodity indices, [...] Read more.
Financial stress can have significant implications for individuals, businesses, asset prices and the economy as a whole. This study examines the nonlinear structure and dynamic changes in the multifractal behavior of cross-correlation between the financial stress index (FSI) and four well-known commodity indices, namely Commodity Research Bureau Index (CRBI), Baltic Dry Index (BDI), London Metal Index (LME) and Brent Oil prices (BROIL), using multifractal detrended cross correlation analysis (MFDCCA). For analysis, we utilized daily values of FSI and commodity index prices from 16 June 2016 to 9 July 2023. The following are the most important empirical findings: (I) All of the chosen commodity market indices show cross correlations with the FSI and have notable multifractal characteristics. (II) The presence of power law cross-correlation implies that a noteworthy shift in FSI is likely to coincide with a considerable shift in the commodity indices. (III) The multifractal cross-correlation is highest between FSI and Brent Oil (BROIL) and lowest with LME. (IV) The rolling windows analysis reveals a varying degree of persistency between FSI and commodity markets. The findings of this study have a number of important implications for commodity market investors and policymakers. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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43 pages, 16277 KiB  
Article
The Multiscale Principle in Nature (Principium luxuriæ): Linking Multiscale Thermodynamics to Living and Non-Living Complex Systems
by Patricio Venegas-Aravena and Enrique G. Cordaro
Fractal Fract. 2024, 8(1), 35; https://doi.org/10.3390/fractalfract8010035 - 4 Jan 2024
Cited by 4 | Viewed by 4335
Abstract
Why do fractals appear in so many domains of science? What is the physical principle that generates them? While it is true that fractals naturally appear in many physical systems, it has so far been impossible to derive them from first physical principles. [...] Read more.
Why do fractals appear in so many domains of science? What is the physical principle that generates them? While it is true that fractals naturally appear in many physical systems, it has so far been impossible to derive them from first physical principles. However, a proposed interpretation could shed light on the inherent principle behind the creation of fractals. This is the multiscale thermodynamic perspective, which states that an increase in external energy could initiate energy transport mechanisms that facilitate the dissipation or release of excess energy at different scales. Within this framework, it is revealed that power law patterns, and to a lesser extent, fractals, can emerge as a geometric manifestation to dissipate energy in response to external forces. In this context, the exponent of these power law patterns (thermodynamic fractal dimension D) serves as an indicator of the balance between entropy production at small and large scales. Thus, when a system is more efficient at releasing excess energy at the microscopic (macroscopic) level, D tends to increase (decrease). While this principle, known as Principium luxuriæ, may sound promising for describing both multiscale and complex systems, there is still uncertainty about its true applicability. Thus, this work explores different physical, astrophysical, sociological, and biological systems to attempt to describe and interpret them through the lens of the Principium luxuriæ. The analyzed physical systems correspond to emergent behaviors, chaos theory, and turbulence. To a lesser extent, the cosmic evolution of the universe and geomorphology are examined. Biological systems such as the geometry of human organs, aging, human brain development and cognition, moral evolution, Natural Selection, and biological death are also analyzed. It is found that these systems can be reinterpreted and described through the thermodynamic fractal dimension. Therefore, it is proposed that the physical principle that could be behind the creation of fractals is the Principium luxuriæ, which can be defined as “Systems that interact with each other can trigger responses at multiple scales as a manner to dissipate the excess energy that comes from this interaction”. That is why this framework has the potential to uncover new discoveries in various fields. For example, it is suggested that the reduction in D in the universe could generate emergent behavior and the proliferation of complexity in numerous fields or the reinterpretation of Natural Selection. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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