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Fractal Fract
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Fractional Dynamics Systems: Modeling, Forecasting, and Control
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Fractional Dynamics Systems: Modeling, Forecasting, and Control

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

Deadline for manuscript submissions: 31 July 2026 | Viewed by 1133

Special Issue Editors

Prof. Dr. Seenith Sivasundaram

E-Mail Website
Guest Editor
Department of Mathematics, Bethune-Cookman University, Daytona Beach, FL 32114, USA
Interests: fractional differential equations; integral boundary conditions; Banach contraction principle; dynamical systems; fractional-order systems; delay differential equations; mathematical modelling; numerical methods; neural networks; optimization; control systems; mathematical medical; biology and environmental sciences
Dr. Devaraj Vivek

E-Mail Website
Guest Editor
Department of Mathematics with Computer Applications, PSG College of Arts & Science, Coimbatore 641014, India
Interests: differential equations; fractional calculus; numerical methods; iterative learning control; impulsive differential equations
Prof. Dr. Carlo Bianca

E-Mail Website
Guest Editor
EFREI Research Laboratory, Université Paris-Panthéon-Assas, 30/32 Avenue de la République, 94800 Villejuif, France
Interests: mathematical modeling and analysis of complex systems; kinetic equations; numerical methods for PDE
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional dynamics systems model systems with "memory" and long-range interdependence using fractional calculus, producing more precise forecasting and control models. They are used to represent intricate phenomena that integer-order models are unable to in a variety of domains, including biology, engineering, and finance. Designing controllers, evaluating system characteristics like stability and controllability, and creating underlying mathematical theories are all included in the study of these systems.

Modeling

  • Memory and Hereditary Properties: Fractional-order models incorporate memory effects, meaning the current state of a system depends not only on its present conditions but also on its entire past history.
  • Non-Local Features: These models are useful for systems with non-local features, such as those with long-range dependencies or effects.
  • Examples: Applications include viscoelastic material modeling, signal processing, financial systems, and the spread of diseases like COVID-19.

Forecasting

  • Accurate Predictions: By accounting for memory, these models can provide more accurate forecasts of future states in complex systems.
  • Data-Driven Models: Techniques like Artificial Neural Networks (ANNs) have been used with fractional models to improve forecasting accuracy in systems like financial markets.
  • Epidemiological Modeling: Fractional-order models have been used to forecast epidemic trends, such as the number of confirmed cases and deaths.

Control

  • Model-Based Control: Researchers develop model-based control design methods specifically for systems described by fractional-order dynamics.
  • Control Design: The control designs can be applied to both fractional- and integer-order controllers, which can then be implemented in real-world applications.
  • Application Example: In one example, a fractional calculus problem was used to design a setpoint filter to control a furnace's temperature, helping to suppress overshooting.
  • System Analysis: Traditional control concepts like stability, controllability, and observability are extended to fractional-order systems, which often have non-standard properties.

Prof. Dr. Seenith Sivasundaram
Dr. Devaraj Vivek
Prof. Dr. Carlo Bianca
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 250 words) can be sent to the Editorial Office for assessment.

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
  • modeling and simulation
  • fractional-order control
  • stability and controllability
  • applications in engineering, biology, and finance
  • non-local dynamics

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Published Papers (2 papers)

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Research

45 pages, 7742 KB  
Article
Fractional-Order Typhoid Fever Dynamics and Parameter Identification via Physics-Informed Neural Networks
by Mallika Arjunan Mani, Kavitha Velusamy, Sowmiya Ramasamy and Seenith Sivasundaram
Fractal Fract. 2026, 10(4), 270; https://doi.org/10.3390/fractalfract10040270 - 21 Apr 2026
Viewed by 330
Abstract
This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order κ∈(0,1]. The population is stratified into five epidemiological classes, namely [...] Read more.
This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order κ∈(0,1]. The population is stratified into five epidemiological classes, namely susceptible (S), asymptomatic (A), symptomatic (I), hospitalised (H), and recovered (R), and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number R0 is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium E0 (when R0≤1) and the endemic equilibrium E* (when R0>1) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of R0 are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network (PINN) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations. PINN embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters Θ = {ϕ,μ,σ,ψ,β} across three fractional orders κ∈{1.0,0.95,0.9}. The estimated parameters show strong agreement with the true values at the classical limit κ=1.0 (MAPE=2.27%), with the natural mortality rate μ recovered with APE≤0.51% and the transmission rate β with APE≤3.63% across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that β can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of 1%, 3%, and 5% demonstrate graceful degradation (MAPE: 0.82%→3.10%→7.31%), confirming the reliability of the proposed framework under realistic observational conditions. Full article
(This article belongs to the Special Issue Fractional Dynamics Systems: Modeling, Forecasting, and Control)
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27 pages, 19923 KB  
Article
Chaotic and Multi-Layer Dynamics in Memristive Fractional Hopfield Neural Networks
by Vignesh Dhakshinamoorthy, Shaobo He and Santo Banerjee
Fractal Fract. 2026, 10(4), 222; https://doi.org/10.3390/fractalfract10040222 - 26 Mar 2026
Viewed by 419
Abstract
Artificial neural network and neuron models have made significant contributions to the area of neurodynamics. Investigating the dynamics of artificial neurons and neural networks is vital in developing brain-like systems and understanding how the brain functions. Neural network models and memristive neurons are [...] Read more.
Artificial neural network and neuron models have made significant contributions to the area of neurodynamics. Investigating the dynamics of artificial neurons and neural networks is vital in developing brain-like systems and understanding how the brain functions. Neural network models and memristive neurons are currently demonstrating a lot of promise in the study of neurodynamics. In order to model the dynamics of biological synapses, this study explores the complex dynamical behavior of a discrete fractional Hopfield-type neural network using a flux-controlled memristive element with periodic memductance. Hyperbolic tangent and sine are the heterogeneous activation functions that are implemented in the proposed system to improve nonlinearity and replicate various forms of brain activity. Stability and bifurcation analyses are used to illustrate the nonlinear dynamical nature of the constructed network model. We examine how the fractional order (ν) and periodical memductance aspects influence the dynamics of the system to emphasize the emerging complex phenomena like multi-layered dynamics and the presence of several distinct dynamical states throughout the system variables. Randomness and complexity of the time series data for the proposed system are illustrated with the help of approximate entropy analysis. These findings could help researchers better understand brain-like memory networks, neuromorphic computers, and the theoretical study of neurological and mental abilities. The study of multi-layer attractors can be useful in advanced sensory devices, neuromorphic devices, and secure communication. Full article
(This article belongs to the Special Issue Fractional Dynamics Systems: Modeling, Forecasting, and Control)
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