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Applied Mathematical Modelling and Dynamical Systems, 3rd Edition
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Applied Mathematical Modelling and Dynamical Systems, 3rd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: 30 September 2026 | Viewed by 1398

Editor

Dr. José Leonel Linhares da Rocha

E-Mail Website
Guest Editor
Department of Mathematics of ISEL—Engineering Superior Institute of Lisbon, Polytechnic Institute of Lisbon, Rua Conselheiro Emídio Navarro 1, 1959-007 Lisbon, Portugal
Interests: discrete dynamical systems; bifurcation theory; population dynamics; topological and metric invariants; complex networks and their applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Building upon the strong foundation and impactful contributions of the first two volumes, we are pleased to announce the opening of the third volume of our Special Issue, titled "Applied Mathematical Modelling and Dynamical Systems, 3rd Edition".

This edition continues to champion the interdisciplinary power of mathematics in bridging theory and application. We invite submissions that push the boundaries of knowledge in dynamical systems and mathematical modelling, with a particular emphasis on novel theoretical frameworks, advanced computational methods, and innovative applications that address complex real-world challenges across science and engineering.

We welcome high-quality original research articles, comprehensive reviews, and concise communications that explore the intricate dynamics of nonlinear, chaotic, and linear systems, and their profound implications in areas such as control theory, network science, population dynamics, synchronization phenomena, and beyond.

Join us in shaping the next chapter of this vibrant collection. We look forward to receiving your valuable contributions.

Prof. Dr. José Leonel Linhares da Rocha
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 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-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • mathematical modelling
  • dynamical systems
  • nonlinear systems
  • theory of singularities
  • fixed point theory
  • bifurcation theory
  • complex systems
  • iteration theory
  • topological dynamics
  • ergodic theory
  • symbolic dynamics
  • population dynamics
  • embedding problems
  • networks
  • synchronization
  • simulation
  • chaos
  • functional equations

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Published Papers (3 papers)

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Research

21 pages, 1295 KB  
Article
Thermal and Mechanical Effects in Thin Lenses Under Ultrafast Laser Heating
by Faizah M. Alharbi and Nafeesa G. Alhendi
Mathematics 2026, 14(13), 2335; https://doi.org/10.3390/math14132335 - 1 Jul 2026
Viewed by 233
Abstract
This study develops a fractional Jeffreys heat conduction model to describe laser-induced thermoelastic distortions in thin optical materials under ultrafast surface heating. The framework employs three fractional parameters to characterize anomalous thermal transport modes: retarded conduction, accelerated conduction, and transitions between super- and [...] Read more.
This study develops a fractional Jeffreys heat conduction model to describe laser-induced thermoelastic distortions in thin optical materials under ultrafast surface heating. The framework employs three fractional parameters to characterize anomalous thermal transport modes: retarded conduction, accelerated conduction, and transitions between super- and sub-diffusive regimes. Thermo-optic effects are represented through a linear relation between temperature and refractive-index perturbation; however, a full optical-aberration decomposition is not claimed in this work. Numerical results demonstrate that anomalous heat transfer significantly affects temperature localization, heat-flux evolution, stress distributions, and OPD-based thermo-optic indicators in components subjected to ultrafast laser pulses. Quantitative optical indicators, including refractive-index variation, optical path difference, wavefront error, focal-length shift, and thermal-lens distortion, are derived from the computed temperature field to connect the thermal solution directly with thin-lens performance. Simulations combining Maple2024 and MATLAB R2023a quantify the coupled thermoelastic-optical response at picosecond time scales. Full article
(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems, 3rd Edition)
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36 pages, 2607 KB  
Article
A Coupled Mathematical Model of Groundwater Dynamics and Salt Transport in a Two-Layer Porous Medium
by Ergashevich Halimjon Khujamatov, Sherzod Daliev, Sherzod Urakov, Sirojiddin Elmonov, Abdinabi Mukhamadiyev and Razvan Craciunescu
Mathematics 2026, 14(10), 1593; https://doi.org/10.3390/math14101593 - 8 May 2026
Viewed by 368
Abstract
Understanding the coupled dynamics of groundwater flow and salinity transport is essential for the sustainable management of aquifer systems, particularly in irrigated and semi-arid regions where evaporation, recharge variability, and groundwater abstraction strongly influence hydrogeological regimes. In multilayer porous media, groundwater-level fluctuations and [...] Read more.
Understanding the coupled dynamics of groundwater flow and salinity transport is essential for the sustainable management of aquifer systems, particularly in irrigated and semi-arid regions where evaporation, recharge variability, and groundwater abstraction strongly influence hydrogeological regimes. In multilayer porous media, groundwater-level fluctuations and salt migration processes are closely interconnected, since hydraulic gradients control solute transport while salinity variations may affect flow behaviour through density-related mechanisms. In this study, a nonlinear mathematical model is developed to describe groundwater-level evolution and salt transport within a two-layer porous medium consisting of a phreatic layer and an underlying confined aquifer. The model accounts for filtration processes, interlayer hydraulic exchange, density-dependent effects, and external forcing factors including surface recharge, evaporation, and pumping. For numerical implementation, the governing equations are discretized using a finite-difference scheme with central spatial approximations and an implicit Crank–Nicolson-type temporal formulation. A hybrid second-order time approximation is introduced for the main-layer equation to improve numerical smoothness and stability. The resulting tridiagonal algebraic systems are solved using the Thomas algorithm within an iterative quasi-linearization framework, ensuring both computational efficiency and numerical robustness. Simulation results reveal a clear difference in the dynamical behaviour of the two layers. The phreatic aquifer exhibits rapid and high-amplitude responses to external forcing, whereas the confined aquifer demonstrates slower and smoother hydraulic and geochemical adjustments. Sensitivity analysis further identifies the filtration coefficient, transmissivity, porosity, density-related parameters, surface flux, and pumping intensity as the dominant factors governing groundwater dynamics and salinity redistribution. The proposed modelling framework provides a reliable tool for analysing coupled groundwater–salinity processes and offers a scientifically grounded basis for groundwater monitoring, salinization risk assessment, and sustainable aquifer management. Full article
(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems, 3rd Edition)
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26 pages, 11053 KB  
Article
Mathematical Modeling and Dynamic Simulation of Frog Jumping for Bio-Inspired Robotics
by Nuria Sánchez Pérez and Juan David Cano-Moreno
Mathematics 2026, 14(9), 1411; https://doi.org/10.3390/math14091411 - 23 Apr 2026
Viewed by 416
Abstract
The biomechanics of frog jumping has been a subject of significant interest in both biology and engineering, driven by the high efficiency of their movement. This study presents the dynamic simulation of a frog’s complete jump cycle, from take-off to landing and re-stabilization, [...] Read more.
The biomechanics of frog jumping has been a subject of significant interest in both biology and engineering, driven by the high efficiency of their movement. This study presents the dynamic simulation of a frog’s complete jump cycle, from take-off to landing and re-stabilization, to advance the development of bio-inspired jumping robots for irregular terrains. As a primary contribution, and unlike previous studies that focus exclusively on the propulsion phase, this work addresses all stages, using direct servomotor actuation without mechanical energy storage. Biological joint kinematics were mathematically characterized using Cubic Smoothing Splines. By empirically tuning the smoothing parameter (p), the trajectories achieved the continuous differentiability required for electromechanical actuation. These curves were implemented into a 3D multibody simulation (Altair Inspire), where a PID-based tracking framework managed the mechanically nonlinear multibody dynamics governing the jump (arising from contact forces, impacts, and time-varying inertial effects) to ensure stabilization during the complex landing phase. Validating the model against previous studies, the simulation successfully achieved a maximum horizontal jump distance of 24.12 cm (4.02 body lengths) and a peak velocity of 1.45 m/s. The kinematic fidelity of the model was mathematically validated, yielding a maximum Normalized Root Mean Square Error (NRMSE) of 4.121% relative to biological reference trajectories. Furthermore, the robustness of the landing and re-stabilization phases was demonstrated through a continuous double jump covering a total distance of 45.83 cm. Finally, a dynamic scaling analysis was performed to evaluate the feasibility of implementing real motors. Ultimately, this study establishes a mathematically robust framework for replicating frog-inspired jumping dynamics, contributing a transferable methodology for the design and control of articulated bio-inspired robotic systems. Full article
(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems, 3rd Edition)
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