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Fractal Fract
Special Issues
Fractional Order Modelling of Dynamical Systems
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Fractional Order Modelling of Dynamical Systems

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

Deadline for manuscript submissions: 31 March 2026 | Viewed by 1954

Special Issue Editor

Dr. Zubair Ahmad

E-Mail Website
Guest Editor
1. Department of Political Science, University of Naples Federico II, 80138 Naples, Italy
2. Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, 81100 Caserta, Italy
Interests: fractional derivatives; mathematical biology; fluid dynamics; dynamical systems

Special Issue Information

Dear Colleagues,

The study of fractional differential equations, which involve derivatives and integrals of non-integer orders, has gained significant momentum in recent years due to their ability to model complex phenomena with memory and hereditary properties. These equations provide a more accurate and flexible framework for describing processes in a wide range of disciplines, including physics, biology, engineering, finance, and materials science.

This Special Issue aims to bring together cutting-edge research in the theory, analysis, numerical methods, and applications of fractional differential equations. We are particularly interested in innovative methodologies and modelling strategies that demonstrate the power of fractional calculus in capturing the dynamics of real-world systems, as well as interdisciplinary studies where fractional models provide new insights or improved results.

We welcome original research articles and review papers that address, but are not limited to, the following topics:

  • Theoretical advances in fractional differential equations;
  • Analytical and numerical methods for solving fractional-order models;
  • Stability, control, and bifurcation analysis of fractional dynamical systems;
  • Applications of fractional models in physics, engineering, biology, and other sciences;
  • Fractional order modelling in fluid mechanics;
  • Fractional-order modelling of viscoelastic, diffusive, and memory-dependent systems;
  • Fractional order models in epidemiology;
  • Computational techniques and software tools for fractional systems;
  • Comparisons of classical and fractional modelling approaches.

We look forward to receiving contributions that will foster progress in this vibrant and interdisciplinary field.

Dr. Zubair Ahmad
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 derivatives
  • numerical methods of fractional order systems
  • fractional dynamical systems
  • stability analysis
  • fractional models in fluid dynamics
  • disease dynamics and epidemiology
  • fractional modelling with applications

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

29 pages, 862 KiB  
Article
Exploring SEIR Influenza Epidemic Model via Fuzzy ABC Fractional Derivatives with Crowley–Martin Incidence Rate
by F. Gassem, Ashraf A. Qurtam, Mohammed Almalahi, Mohammed Rabih, Khaled Aldwoah, Abdelaziz El-Sayed and E. I. Hassan
Fractal Fract. 2025, 9(7), 402; https://doi.org/10.3390/fractalfract9070402 - 23 Jun 2025
Viewed by 349
Abstract
Despite initial changes in respiratory illness epidemiology due to SARS-CoV-2, influenza activity has returned to pre-pandemic levels, highlighting its ongoing challenges. This paper investigates an influenza epidemic model using a Susceptible-Exposed-Infected-Recovered (SEIR) framework, extended with fuzzy Atangana–Baleanu–Caputo (ABC) fractional derivatives to incorporate uncertainty [...] Read more.
Despite initial changes in respiratory illness epidemiology due to SARS-CoV-2, influenza activity has returned to pre-pandemic levels, highlighting its ongoing challenges. This paper investigates an influenza epidemic model using a Susceptible-Exposed-Infected-Recovered (SEIR) framework, extended with fuzzy Atangana–Baleanu–Caputo (ABC) fractional derivatives to incorporate uncertainty (via fuzzy numbers for state variables) and memory effects (via the ABC fractional derivative for non-local dynamics). We establish the theoretical foundation by defining the fuzzy ABC derivatives and integrals based on the generalized Hukuhara difference. The existence and uniqueness of the solutions for the fuzzy fractional SEIR model are rigorously proven using fixed-point theorems. Furthermore, we analyze the system’s disease-free and endemic equilibrium points under the fractional framework. A numerical scheme based on the fractional Adams–Bashforth method is used to approximate the fuzzy solutions, providing interval-valued results for different uncertainty levels. The study demonstrates the utility of fuzzy fractional calculus in providing a more flexible and potentially realistic approach to modeling epidemic dynamics under uncertainty. Full article
(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)
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38 pages, 1459 KiB  
Article
A Comparative Analysis of Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin Incidence Rates of a Piecewise Dengue Fever Dynamics Model
by Faten H. Damag, Ashraf A. Qurtam, Mohammed Almalahi, Khaled Aldwoah, Mohamed Adel, Alaa M. Abd El-Latif and E. I. Hassan
Fractal Fract. 2025, 9(7), 400; https://doi.org/10.3390/fractalfract9070400 - 22 Jun 2025
Viewed by 205
Abstract
Dengue fever remains a major global health threat, and mathematical models are crucial for predicting its spread and evaluating control strategies. This study introduces a highly flexible dengue transmission model using a novel piecewise fractional derivative framework, which can capture abrupt changes in [...] Read more.
Dengue fever remains a major global health threat, and mathematical models are crucial for predicting its spread and evaluating control strategies. This study introduces a highly flexible dengue transmission model using a novel piecewise fractional derivative framework, which can capture abrupt changes in epidemic dynamics, such as those caused by public health interventions or seasonal shifts. We conduct a rigorous comparative analysis of four widely used but distinct mechanisms of disease transmission (incidence rates): Harmonic Mean, Holling Type II, Beddington–DeAngelis, and Crowley–Martin. The model’s well-posedness is established, and the basic reproduction number (ℜ0) is derived for each incidence function. Our central finding is that the choice of this mathematical mechanism critically alters predictions. For example, models that account for behavioral changes (Beddington–DeAngelis, Crowley–Martin) identify different key drivers of transmission compared to simpler models. Sensitivity analysis reveals that vector mortality is the most influential control parameter in these more realistic models. These results underscore that accurately representing transmission behavior is essential for reliable epidemic forecasting and for designing effective, targeted intervention strategies. Full article
(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)
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20 pages, 386 KiB  
Article
Some Fixed Point Results for Novel Contractions with Applications in Fractional Differential Equations for Market Equilibrium and Economic Growth
by Min Wang, Muhammad Din and Mi Zhou
Fractal Fract. 2025, 9(5), 324; https://doi.org/10.3390/fractalfract9050324 - 19 May 2025
Viewed by 354
Abstract
In this study, we introduce two new classes of contractions, namely enriched (I,ρ,χ)-contractions and generalized enriched (I,ρ,χ)-contractions, within the context of normed spaces. These classes generalize several well-known contraction [...] Read more.
In this study, we introduce two new classes of contractions, namely enriched (I,ρ,χ)-contractions and generalized enriched (I,ρ,χ)-contractions, within the context of normed spaces. These classes generalize several well-known contraction types, including χ-contractions, Banach contractions, enriched contractions, Kannan contractions, Bianchini contractions, Zamfirescu contractions, non-expansive mappings, and (ρ,χ)-enriched contractions. We establish related fixed point results for the novel contractions in normed spaces endowed with the binary relations preserving key symmetric properties, ensuring consistency and applicability. The Krasnoselskij iteration method is refined to incorporate symmetric constraints, facilitating fixed point identification within these spaces. By appropriately selecting constants in the definition of enriched (I,ρ,χ)-contractions, employing a suitable binary relation, or control function χ∈Θ, our framework generalizes and extends classical fixed point theorems. Illustrative examples highlight the significance of our findings in reinforcing fixed point conditions and demonstrating their broader applicability. Additionally, this paper explores how these ideas guarantee the stability of the production–consumption markets equilibrium and the economic growth model. Full article
(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)
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