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Advances in Dynamical Systems, Differential Equations, and Their Applications
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Advances in Dynamical Systems, Differential Equations, and Their Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 31 August 2025 | Viewed by 7526

Special Issue Editors

Dr. Carla M. A. Pinto

E-Mail Website
Guest Editor
Instituto Superior de Engenharia do Porto, Rua Dr. António Bernardino de Almeida, 431 4249-015 Porto, Portugal
Interests: epidemiological and within-host models; dynamical systems; fractional order models
Special Issues, Collections and Topics in MDPI journals
Dr. Helena Reis

E-Mail Website
Guest Editor
Faculdade de Economia, Universidade do Porto, 4200-464 Porto, Portugal
Interests: dynamical systems; manifolds; lie algebras; holomorphic flows; geometry

Special Issue Information

Dear Colleagues,

We cordially invite you to submit articles for a Special Issue of Mathematics on dynamical systems and differential equations. The title of this Special Issue not only reflects the topicality of the Special Issue itself but also alludes to the upcoming International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMASC’24). ICMASC’24 is a peer-reviewed conference that highlights various aspects of mathematical analysis and its applications in science and engineering.

The focus of this Special Issue will revolve around the broad spectrum of dynamical systems, encompassing iterative dynamics, ordinary differential equations, and (evolutionary) partial differential equations. We encourage submissions addressing these topics either from a theoretical perspective or exploring their diverse applications in physics (such as in cosmology, quantum physics, matter theory, thermodynamics) or in conventional fields like control theory, artificial intelligence, diagnosis algorithms, among others. We want to stress that this Special Issue is also open to submissions from authors who are interested in the topic but did not attend the conference.

We kindly invite the submission of both original research works and exceptional review articles for consideration in this Special Issue. 

Dr. Carla M. A. Pinto
Dr. Helena Reis
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. 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

  • iterative dynamics
  • ordinary differential equations
  • (evolutionary) partial differential equations
  • applications to physics (cosmology, quantum physics, matter theory, thermodynamics) and other sciences

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Published Papers (5 papers)

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Research

12 pages, 1392 KiB  
Article
The Global Dynamics of the Painlevé–Gambier Equations XVIII, XXI, and XXII
by Jie Li and Jaume Llibre
Mathematics 2025, 13(5), 756; https://doi.org/10.3390/math13050756 - 25 Feb 2025
Viewed by 511
Abstract
In this paper, we describe the global dynamics of the Painlevé–Gambier equations numbered XVIII: x″−(x′)2/(2x)−4x2=0, XXI: [...] Read more.
In this paper, we describe the global dynamics of the Painlevé–Gambier equations numbered XVIII: x″−(x′)2/(2x)−4x2=0, XXI: x″−3(x′)2/(4x)−3x2, and XXII: x″−3(x′)2/(4x)+1=0. We obtain three rational functions as their first integrals and classify their phase portraits in the Poincaré disc. The main reason for considering these three Painlevé–Gambier equations is due to the paper of Guha, P., et al., where the authors studied these three differential equations in order to illustrate a method to generate nonlocal constants of motion for a special class of nonlinear differential equations. Here, we want to complete their studies describing all of the dynamics of these equations. This demonstrates that the phase portraits of equations XVIII and XXI in the Poincaré disc are topologically equivalent. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems, Differential Equations, and Their Applications)
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20 pages, 581 KiB  
Article
Mathematical Modelling of Viscoelastic Media Without Bulk Relaxation via Fractional Calculus Approach
by Marina V. Shitikova and Konstantin A. Modestov
Mathematics 2025, 13(3), 350; https://doi.org/10.3390/math13030350 - 22 Jan 2025
Viewed by 709
Abstract
In the present paper, several viscoelastic models are studied for cases when time-dependent viscoelastic operators of Lamé’s parameters are represented in terms of the fractional derivative Kelvin–Voigt, Scott-Blair, Maxwell, and standard linear solid models. This is practically important since precisely these parameters define [...] Read more.
In the present paper, several viscoelastic models are studied for cases when time-dependent viscoelastic operators of Lamé’s parameters are represented in terms of the fractional derivative Kelvin–Voigt, Scott-Blair, Maxwell, and standard linear solid models. This is practically important since precisely these parameters define the velocities of longitudinal and transverse waves propagating in three-dimensional media. Using the algebra of dimensionless Rabotnov’s fractional exponential functions, time-dependent operators for Poisson’s ratios have been obtained and analysed. It is shown that materials described by some of such models are viscoelastic auxetics because the Poisson’s ratios of such materials are time-dependent operators which could take on both positive and negative magnitudes. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems, Differential Equations, and Their Applications)
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25 pages, 545 KiB  
Article
Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species
by Na Shi, Xin Wu and Zhaohai Ma
Mathematics 2025, 13(2), 197; https://doi.org/10.3390/math13020197 - 9 Jan 2025
Viewed by 556
Abstract
We investigate the multidimensional stability of planar traveling waves in competitive–cooperative Lotka–Volterra system of three species in n-dimensional space. For planar traveling waves with speed c>c*, we establish their exponential stability in L∞(Rn) [...] Read more.
We investigate the multidimensional stability of planar traveling waves in competitive–cooperative Lotka–Volterra system of three species in n-dimensional space. For planar traveling waves with speed c>c*, we establish their exponential stability in L∞(Rn), which is expressed as t−n2e−ετσt, where σ>0 is a constant and ετ∈(0,1) depends on the time delay τ>0 as a decreasing function ετ=ε(τ). The time delay is shown to significantly reduce the decay rate of the solution. Additionally, for planar traveling waves with speed c=c*, we demonstrate their algebraic stability in the form t−n2. Our analysis employs the Fourier transform and a weighted energy method with a carefully chosen weight function. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems, Differential Equations, and Their Applications)
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23 pages, 2198 KiB  
Article
Dynamics of Some Perturbed Morse-Type Oscillators: Simulations and Applications
by Nikolay Kyurkchiev, Tsvetelin Zaevski, Anton Iliev, Todor Branzov, Vesselin Kyurkchiev and Asen Rahnev
Mathematics 2024, 12(21), 3368; https://doi.org/10.3390/math12213368 - 27 Oct 2024
Cited by 4 | Viewed by 1000
Abstract
The purpose of this paper is to investigate some Morse-type oscillators. In its original form, it is a model for describing the vibrations of a diatomic molecule. The Morse potential generalizes the harmonic oscillator by introducing deviations from the classical theoretical model. In [...] Read more.
The purpose of this paper is to investigate some Morse-type oscillators. In its original form, it is a model for describing the vibrations of a diatomic molecule. The Morse potential generalizes the harmonic oscillator by introducing deviations from the classical theoretical model. In the present study, we perturbed the Morse differential equation by several periodic terms based on the cosine function and by a damping term. The frequency is driven by different coefficients. The size of the deviations is controlled by another constant. We provide two modifications w.r.t. the damping term. The Melnikov approach is applied as an indicator of the possible chaotic opportunities. We also propose a novel approach for stochastic control of the perturbations. It is based on the assumption that the coefficients of the periodic terms are the probabilities of underlying distribution. As a result, the dynamics are driven by its characteristic function. Several applications are considered. We demonstrate some specialized modules for investigating the dynamics of the proposed models, along with the synthesis of radiating antenna patterns. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems, Differential Equations, and Their Applications)
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45 pages, 4025 KiB  
Article
Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models
by Reinhard Schlickeiser and Martin Kröger
Mathematics 2024, 12(7), 941; https://doi.org/10.3390/math12070941 - 22 Mar 2024
Cited by 6 | Viewed by 2384
Abstract
The susceptible–infected–recovered–vaccinated–deceased (SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between [...] Read more.
The susceptible–infected–recovered–vaccinated–deceased (SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between different fractions. We investigate a special class of exact solutions and accurate analytical approximations for the SIRVD and SIRD compartment models. While the SIRVD and SIRD equations pose complex integro-differential equations for the rate of new infections and the fractions as a function of time, a simpler approach considers determining equations for the sum of ratios for given variations. This approach enables us to derive fully exact analytical solutions for the SIRVD and SIRD models. For nonlinear models with a high-dimensional parameter space, such as the SIRVD and SIRD models, analytical solutions, exact or accurately approximative, are of high importance and interest, not only as suitable benchmarks for numerical codes, but especially as they allow us to understand the critical behavior of epidemic outbursts as well as the decisive role of certain parameters. In the second part of our study, we apply a recently developed analytical approximation for the SIR and SIRV models to the more general SIRVD model. This approximation offers accurate analytical expressions for epidemic quantities, such as the rate of new infections and the fraction of infected persons, particularly when the cumulative fraction of infections is small. The distinction between recovered and deceased individuals in the SIRVD model affects the calculation of the death rate, which is proportional to the infected fraction in the SIRVD/SIRD cases but often proportional to the rate of new infections in many SIR models using an a posteriori approach. We demonstrate that the temporal dependence of the infected fraction and the rate of new infections differs when considering the effects of vaccinations and when the real-time dependence of fatality and recovery rates diverge. These differences are highlighted for stationary ratios and gradually decreasing fatality rates. The case of stationary ratios allows one to construct a new powerful diagnostics method to extract analytically all SIRVD model parameters from measured COVID-19 data of a completed pandemic wave. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems, Differential Equations, and Their Applications)
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