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Fractal Fract
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Fractional Behavior in Nature 2019
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Fractional Behavior in Nature 2019

A special issue of Fractal and Fractional (ISSN 2504-3110).

Deadline for manuscript submissions: closed (30 November 2020) | Viewed by 21219

Special Issue Editor

Dr. Manuel Duarte Ortigueira

E-Mail Website
Guest Editor
Centre of Technology and Systems-UNINOVA, NOVA School of Science and Technology, NOVA University of Lisbon, Quinta da Torre, 2829-516 Caparica, Portugal
Interests: signal processing; fractional signals and systems; EEG and ECG processing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is already known that the non-integer order systems can describe the dynamical behavior of materials and processes over vast time and frequency scales, with very concise and computable models.

  1. There is evidence that most of the biological signals have spectra that do not increase or decrease by multiples of 20 dB/sec.
  2. The long-range processes (1/f noise sources)—the fractional Brownian motion (fBm) is the most famous—are very common in nature.
  3. The power law behavior can be found in many processes.

On the other hand, and looking from a much deeper perspective, the fractional derivative implies causality. By respecting the proper time order and including the effects of the past on the evolution of systems and processes, we open the door to a more realistic, non-Markovian view of the world, without drastically increasing the complexity of the system descriptions.

Prof. Dr. Manuel Ortigueira
Guest Editor

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 derivative
  • fractional integral
  • long range
  • power law
  • fractional models
  • fractional discrete-time systems
  • fractional continuous-time systems
  • ARFIMA

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Published Papers (8 papers)

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Editorial

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2 pages, 168 KiB  
Editorial
Editorial for Special Issue “Fractional Behavior in Nature 2019”
by Manuel Duarte Ortigueira
Fractal Fract. 2021, 5(4), 186; https://doi.org/10.3390/fractalfract5040186 - 26 Oct 2021
Cited by 1 | Viewed by 1221
Abstract
The presence of fractional behavior in nature is unquestionable [...] Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

Research

Jump to: Editorial

17 pages, 917 KiB  
Article
Signal Propagation in Electromagnetic Media Modelled by the Two-Sided Fractional Derivative
by Jacek Gulgowski, Dariusz Kwiatkowski and Tomasz P. Stefański
Fractal Fract. 2021, 5(1), 10; https://doi.org/10.3390/fractalfract5010010 - 18 Jan 2021
Cited by 8 | Viewed by 2147
Abstract
In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its [...] Read more.
In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality of the system is investigated; its output is an electromagnetic signal propagating in media described by the time-domain two-sided fractional derivative. It is demonstrated that, for the derivative order in the range [1,+∞), the transfer function describing attenuated signal propagation is not causal for any value of the asymmetry parameter of the derivative. On the other hand, it is shown that, for derivative orders in the range (0,1), the transfer function is causal if and only if the asymmetry parameter is equal to certain specific values corresponding to the left-sided Grünwald–Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers–Krönig relations for logarithm of the transfer function are presented as well. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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29 pages, 554 KiB  
Article
Biased Continuous-Time Random Walks with Mittag-Leffler Jumps
by Thomas M. Michelitsch, Federico Polito and Alejandro P. Riascos
Fractal Fract. 2020, 4(4), 51; https://doi.org/10.3390/fractalfract4040051 - 31 Oct 2020
Cited by 13 | Viewed by 2502
Abstract
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we [...] Read more.
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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15 pages, 6290 KiB  
Article
Fractal and Fractional Derivative Modelling of Material Phase Change
by Harry Esmonde
Fractal Fract. 2020, 4(3), 46; https://doi.org/10.3390/fractalfract4030046 - 14 Sep 2020
Cited by 8 | Viewed by 2416
Abstract
An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations [...] Read more.
An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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18 pages, 554 KiB  
Article
Fractional SIS Epidemic Models
by Caterina Balzotti, Mirko D’Ovidio and Paola Loreti
Fractal Fract. 2020, 4(3), 44; https://doi.org/10.3390/fractalfract4030044 - 31 Aug 2020
Cited by 20 | Viewed by 3697
Abstract
In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with [...] Read more.
In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order α converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the α-SIS models. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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15 pages, 474 KiB  
Article
Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective
by Agneta M. Balint and Stefan Balint
Fractal Fract. 2020, 4(3), 36; https://doi.org/10.3390/fractalfract4030036 - 21 Jul 2020
Cited by 13 | Viewed by 3017
Abstract
In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of [...] Read more.
In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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13 pages, 285 KiB  
Article
Fractional Derivatives and Dynamical Systems in Material Instability
by Peter B. Béda
Fractal Fract. 2020, 4(2), 14; https://doi.org/10.3390/fractalfract4020014 - 16 Apr 2020
Cited by 2 | Viewed by 2271
Abstract
Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a [...] Read more.
Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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12 pages, 3713 KiB  
Article
Fractional-Order Models for Biochemical Processes
by Eva-H. Dulf, Dan C. Vodnar, Alex Danku, Cristina-I. Muresan and Ovidiu Crisan
Fractal Fract. 2020, 4(2), 12; https://doi.org/10.3390/fractalfract4020012 - 10 Apr 2020
Cited by 12 | Viewed by 2989
Abstract
Biochemical processes present complex mechanisms and can be described by various computational models. Complex systems present a variety of problems, especially the loss of intuitive understanding. The present work uses fractional-order calculus to obtain mathematical models for erythritol and mannitol synthesis. The obtained [...] Read more.
Biochemical processes present complex mechanisms and can be described by various computational models. Complex systems present a variety of problems, especially the loss of intuitive understanding. The present work uses fractional-order calculus to obtain mathematical models for erythritol and mannitol synthesis. The obtained models are useful for both prediction and process optimization. The models present the complex behavior of the process due to the fractional order, without losing the physical meaning of gain and time constants. To validate each obtained model, the simulation results were compared with experimental data. In order to highlight the advantages of fractional-order models, comparisons with the corresponding integer-order models are presented. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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