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

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 26308

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Dr. João Manuel Gonçalves Cabral

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Faculdade de Ciências e Tecnologia, University of Azores, 9500-321 Ponta Delgada, Portugal
Interests: rational maps iteration; dynamical systems; mathematical modeling; environment dynamics of mathematics methodology and teaching
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Prof. Dr. Daniele Fournier-Prunaret

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Institut National des Sciences Appliquées de Toulouse, 31400 Toulouse, France
Interests: nonlinear systems; complex behaviours; bifurcations; chaos
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Prof. Dr. José Leonel Linhares da Rocha

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CEAUL and Department of Mathematics of ISEL-Engineering Superior Institute of Lisbon, Polytechnic Institute of Lisbon, Rua Conselheiro Emídio Navarro 1, 1959-007 Lisboa, Portugal
Interests: nonlinear dynamics; population dynamics (Allee effects); bifurcation theory; networks; synchronization and applications
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Special Issue Information

Dear Colleagues,

We are pleased to announce a Special Issue of the journal Mathematics entitled “Applied Mathematical Modelling and Dynamical Systems”. Many problems in our society are solved several times through the construction of little bridges connecting different branches of Mathematics. From the most theoretical aspect to the most practical equation, from the most complex nonlinear system to the most iterative process, from the most chaotical event to the most linear, great and incredible phenomena, sometimes missed when studied alone with the tools of an unique field, can be spotted when making connections between several fields, usually giving birth to amazing theories, which, when applied, produce results with the power of make human life better.

Dynamical systems can model many different phenomena in nature and society. The goal of all theoretical studies is to analyze and understand processes often related to different fields of application such as control theory, bifurcation theory, population dynamics, networks, synchronization phenomena, electronics, physics, mechanics, economics, biology, and ecology, among so many others.

This Special Issue welcomes original research articles, short communications, and review papers. Potential topics include theoretical studies as well as analyses of applied models related to all the fields cited above.

Prof. Dr. João Cabral
Prof. Dr. Daniele Fournier-Prunaret
Prof. Dr. José Leonel Linhares da Rocha
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

mathematical modeling
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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Related Special Issue

Applied Mathematical Modelling and Dynamical Systems, 2nd Edition in Mathematics (9 articles)

Published Papers (12 papers)

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Research

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20 pages, 726 KiB

Open AccessArticle

Mathematical Modeling of Toxoplasmosis in Cats with Two Time Delays under Environmental Effects

by Sharmin Sultana, Gilberto González-Parra and Abraham J. Arenas

Mathematics 2023, 11(16), 3463; https://doi.org/10.3390/math11163463 - 10 Aug 2023

Cited by 1 | Viewed by 1620

Abstract

In this paper, we construct a more realistic mathematical model to study toxoplasmosis dynamics. The model considers two discrete time delays. The first delay is related to the latent phase, which is the time lag between when a susceptible cat has effective contact with an oocyst and when it begins to produce oocysts. The second discrete time delay is the time that elapses from when the oocysts become present in the environment to when they are able to infect. The main aim in this paper is to find the conditions under which the toxoplasmosis can disappear from the cat population and to study whether the time delays can affect the qualitative properties of the model. Thus, we investigate the impact of the combination of two discrete time delays on the toxoplasmosis dynamics. Using dynamical systems theory, we are able to find the basic reproduction number

R_{0}^{d}

that determines the global long-term dynamics of the toxoplasmosis. We prove that, if

R_{0}^{d} < 1

, the toxoplasmosis will be eradicated and that the toxoplasmosis-free equilibrium is globally stable. We design a Lyapunov function in order to prove the global stability of the toxoplasmosis-free equilibrium. We also prove that, if the threshold parameter

R_{0}^{d}

is greater than one, then there is only one toxoplasmosis-endemic equilibrium point, but the stability of this point is not theoretically proven. However, we obtained partial theoretical results and performed numerical simulations that suggest that, if

R_{0}^{d} > 1

, then the toxoplasmosis-endemic equilibrium point is globally stable. In addition, other numerical simulations were performed in order to help to support the theoretical stability results. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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9 pages, 259 KiB

Open AccessArticle

Distributional Chaos and Sensitivity for a Class of Cyclic Permutation Maps

by Yu Zhao, Waseem Anwar, Risong Li, Tianxiu Lu and Zhiwen Mo

Mathematics 2023, 11(15), 3310; https://doi.org/10.3390/math11153310 - 27 Jul 2023

Cited by 1 | Viewed by 833

Abstract

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form [...] Read more.

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form

φ (b_{1}, b_{2}, \dots, b_{p}) = (u_{p} (b_{p}), u_{1} (b_{1}), \dots, u_{p - 1} (b_{p - 1})),

where

b_{j} \in H_{j}

(

j \in {1, 2, \dots, p}

p \geq 2

is an integer, and

H_{j}

(

j \in {1, 2, \dots, p}

) are compact subintervals of the real line

R = (- \infty, + \infty)

u_{j} : H_{j} \to H_{j + 1} (j = 1, 2, \dots, p - 1)

and

u_{p} : H_{p} \to H_{1}

are continuous maps. Necessary and sufficient conditions for a class of cyclic permutation maps to have Li–Yorke chaos, distributional chaos in a sequence, distributional chaos, or Li–Yorke sensitivity are given. These results extend the existing ones. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

18 pages, 1798 KiB

Open AccessArticle

Influence Maximization Dynamics and Topological Order on Erdös-Rényi Networks

by J. Leonel Rocha, Sónia Carvalho, Beatriz Coimbra, Inês Henriques and Juliana Pereira

Mathematics 2023, 11(15), 3299; https://doi.org/10.3390/math11153299 - 26 Jul 2023

Cited by 2 | Viewed by 1310

Abstract

This paper concerns the study of the linear threshold model in random networks, specifically in Erdös-Rényi networks. In our approach, we consider an activation threshold defined by the expected value for the node degree and the associated influence activation mapping. According to these assumptions, we present a theoretical procedure for the linear threshold model, under fairly general conditions, regarding the topological structure of the networks and the activation threshold. Aiming at the dynamics of the influence maximization process, we analyze and discuss different choices for the seed set based on several centrality measures along with the state conditions for the procedure to trigger. The topological entropy established for Erdös-Rényi networks defines a topological order for this type of random networks. Sufficient conditions are presented for this topological entropy to be characterized by the spectral radius of the associated adjacency matrices. Consequently, a number of properties are proved. The threshold dynamics are analyzed through the relationship between the activation threshold and the topological entropy. Numerical studies are included to illustrate the theoretical results. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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15 pages, 847 KiB

Open AccessArticle

Revealing Chaos Synchronization Below the Threshold in Coupled Mackey–Glass Systems

by Marat Akhmet, Kağan Başkan and Cihan Yeşil

Mathematics 2023, 11(14), 3197; https://doi.org/10.3390/math11143197 - 21 Jul 2023

Cited by 4 | Viewed by 1321

Abstract

This study presents a novel concept in chaos synchronization, delta synchronization of chaos, which reveals the presence of chaotic models evolving in unison even in the absence of generalized synchronization. Building upon an analysis of unpredictability in Poincaré chaos, we apply this approach to unilaterally coupled time-delay Mackey–Glass models. The main novelty of our investigation lies in unveiling the synchronization phenomenon for a coupling constant below the synchronization threshold, an unattainable domain for conservative methods. Furthermore, we rigorously examine the coexistence of generalized synchronization and complete synchronization of unpredictability, which is a special case of delta synchronization, above the threshold. Therefore, the threshold is no longer a requirement for the synchronization of chaos in view of the present results. Additionally, transitions to fully chaotic regimes are demonstrated via return maps, phase portraits, and a bifurcation diagram. The findings are substantiated by tables and novel numerical characteristics. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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13 pages, 362 KiB

Open AccessArticle

Attractors in Pattern Iterations of Flat Top Tent Maps

by Luis Silva

Mathematics 2023, 11(12), 2677; https://doi.org/10.3390/math11122677 - 13 Jun 2023

Viewed by 1050

Abstract

Flat-topped one-dimensional maps have been used in the control of chaos in one-dimensional dynamical systems. In these applications, this mechanism is known as simple limiter control. In this paper, we will consider the introduction of simple limiters u in the tent map, according to a time-dependent scheme defined by a binary sequence s, the iteration pattern. We will define local and Milnor attractors in this non-autonomous context and study the dependence of their existence and coexistence on the value of the limiter u and on the pattern s. Using symbolic dynamics, we will be able to characterize the families of pairs

(u, s)

for which these attractors exist and coexist, as well as fully describe them. We will observe that this non-autonomous context provides a richness of behaviors that are not possible in the autonomous case. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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18 pages, 314 KiB

Open AccessArticle

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete-Time Systems Using Optimization Approaches

by Yuyao Lei, Hongli Yang and Ivan Ganchev Ivanov

Mathematics 2023, 11(11), 2419; https://doi.org/10.3390/math11112419 - 23 May 2023

Cited by 1 | Viewed by 1114

Abstract

A positively invariant set is an important concept in dynamical systems. The study of positively invariant set conditions for discrete-time systems is one interesting topic in both theoretical studies and practical applications research. Different methods for characterizing the invariance of different types of sets have been established. For example, the ellipsoidal and the Lorenz cone, which are quadratic convex sets, have different properties from a polyhedral set. This paper presents an optimization method and a dual optimization method to characterize the positive invariance of the ellipsoidal and the Lorenz cone. The proposed methods are applicable to both linear and nonlinear discrete-time systems. Using nonlinear programming and an induced norm, the positive invariance condition problems are transformed into optimization problems, and the dual optimization method is also used to give equivalent dual forms. Fewer results on the positive invariance condition of Lorenz cones can be found than for the other type of set; this paper fulfills the results of this problem. In addition, the proposed methods in this paper provide more options for checking the positive invariance of quadratic convex sets from the perspective of optimization and dual optimization. The effectiveness of this method is demonstrated by numerical examples. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

13 pages, 443 KiB

Open AccessArticle

Lie Symmetry Analysis and Conservation Laws of the Axially Loaded Euler Beam

by Lili Xia and Xinsheng Ge

Mathematics 2022, 10(15), 2759; https://doi.org/10.3390/math10152759 - 3 Aug 2022

Cited by 3 | Viewed by 1674

Abstract

By applying the Lie symmetry method, group-invariant solutions are constructed for axially loaded Euler beams. The corresponding mathematical models of the beams are formulated. After introducing the infinitesimal transformations, the determining equations of Lie symmetry are proposed via Lie point transformations acting on the original equations. The infinitesimal generators of symmetries of the systems are presented with Maple. The corresponding vector fields are given to span the subalgebra of the systems. Conserved vectors are derived by using two methods, namely, the multipliers method and Noether’s theorem. Noether conserved quantities are obtained using the structure equation, satisfied by the gauge functions. The fluxes of the conservation laws could also be proposed with the multipliers. The relations between them are discussed. Furthermore, the original equations of the systems could be transformed into ODEs and the exact explicit solutions are provided. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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15 pages, 29218 KiB

Open AccessEditor’s ChoiceArticle

Coexisting Attractor in a Gyrostat Chaotic System via Basin of Attraction and Synchronization of Two Nonidentical Mechanical Systems

by Muhammad Marwan, Vagner Dos Santos, Muhammad Zainul Abidin and Anda Xiong

Mathematics 2022, 10(11), 1914; https://doi.org/10.3390/math10111914 - 2 Jun 2022

Cited by 11 | Viewed by 2935

Abstract

This paper is divided into two main portions. First, we look at basins of attraction as a tool with a unique set of characteristics for discussing multistability and coexisting attractors in a gyrostat chaotic system. For the validation of coexisting attractors in different basins, several approaches such as bifurcation diagrams, Lyapunov exponents, and the Poincaré section are applied. The second half of the study synchronizes two mechanical chaotic systems using a novel controller, with gyrostat and quadrotor unmanned aerial vehicle (QUAV) chaotic systems acting as master and slave systems, respectively. The error dynamical system and the parameter updated law are built using Lyapunov’s theory, and it is discovered that under certain parametric conditions, the trajectories of the QUAV chaotic system overlap and begin to match the features of the gyrostat chaotic system. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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21 pages, 4609 KiB

Open AccessArticle

Mathematical Modeling of the Operation of an Expander-Generator Pressure Regulator in Non-Stationary Conditions of Small Gas Pressure Reduction Stations

by Artem Evgenevich Belousov and Egor Sergeevich Ovchinnikov

Mathematics 2022, 10(3), 393; https://doi.org/10.3390/math10030393 - 27 Jan 2022

Cited by 8 | Viewed by 3493

Abstract

Long-distance gas transfer requires high pressure, which has to be reduced before the gas is conveyed to the customers. This pressure reduction takes place at natural gas pressure reduction stations, where gas pressure is decreased by using gas flow energy for overcoming local resistance, represented by a throttling valve. This pressure energy can be reused, but it is difficult to implement it at small pressure reduction stations, as the values of unsteadiness significantly increase when the gas approaches consumers, whereas gas flow rate and pressure decrease. This work suggests replacing throttling valves at small pressure reduction stations for expander-generator units, based on volumetric expanders. Two implementations are proposed. A mathematical model of gas-dynamic processes, which take place in expander-generator units, was developed using math equations. With its help, a comparison was made of the stability of the operation of two possible control schemes in non-stationary conditions, and the feasibility of using an expander-generator regulator as a primary one for a small natural gas pressure reduction station was confirmed. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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15 pages, 5484 KiB

Open AccessArticle

Modelling of Heat Transfer Processes in Heat Exchangers for Cardiopulmonary Bypass

by Valentyna Danilova, Vladyslav Shlykov, Vitalii Kotovskyi, Nikolaj Višniakov and Andžela Šešok

Mathematics 2021, 9(23), 3125; https://doi.org/10.3390/math9233125 - 4 Dec 2021

Viewed by 2992

Abstract

A model of the heat exchange process in the heat exchanger of the cardiopulmonary bypass device is proposed which allows for automation of the process of temperature regulation in the cardiopulmonary bypass with an accuracy of ±1 °C during cardiac surgery under controlled cooling and warming of the patient’s heart and brain. The purpose of this research is to create a concept and model of the temperature control circuit using the MSC Easy5 system, the creation of mathematical models of blocks of the temperature control circuit, and the description of the principle of temperature control in the cardiopulmonary bypass circuit. The model of the temperature control loop in the heat exchanger of the heart-lung machine was created using the MSC Easy5 system with a programmable microcontroller. The microcontroller implements a specialized temperature control algorithm in the C language. The model allows the creation of a full-fledged virtual prototype of a temperature control device in a heat exchanger, and helps to conduct virtual tests of the developed device at the design stage. The model identifies control system flaws and influences decisions made before producing an official prototype of the product. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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13 pages, 2825 KiB

Open AccessArticle

Symmetry Preserving Discretization of the Hamiltonian Systems with Holonomic Constraints

by Lili Xia, Mengmeng Wu and Xinsheng Ge

Mathematics 2021, 9(22), 2959; https://doi.org/10.3390/math9222959 - 19 Nov 2021

Viewed by 1894

Abstract

Symmetry preserving difference schemes approximating equations of Hamiltonian systems are presented in this paper. For holonomic systems in the Hamiltonian framework, the symmetrical operators are obtained by solving the determining equations of Lie symmetry with the Maple procedure. The difference type of symmetry preserving invariants are constructed based on the three points of the lattice and the characteristic equations. The difference scheme is constructed by using these discrete invariants. An example is presented to illustrate the applications of the results. The solutions of the invariant numerical schemes are compared to the noninvariant ones, the standard and the exact solutions. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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Review

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17 pages, 1818 KiB

Open AccessReview

A Mathematical Solution to the Computational Fluid Dynamics (CFD) Dilemma

by Stefan Heinz

Mathematics 2023, 11(14), 3199; https://doi.org/10.3390/math11143199 - 21 Jul 2023

Cited by 9 | Viewed by 3317

Abstract

Turbulent flows of practical relevance are often characterized by high Reynolds numbers and solid boundaries. The need to account for flow separation seen in such flows requires the use of (partially) resolving simulation methods on relatively coarse grids. The development of such computational methods is characterized by stagnation. Basically, only a few methods are regularly applied that are known to suffer from significant shortcomings: such methods are often characterized by the significant uncertainty of the predictions due to a variety of adjustable simulation settings, their computational cost can be essential because performance shortcomings need to be compensated by a higher resolution, and there are questions about their reliability because the flow resolving ability is unclear; hence, all such predictions require justification. A substantial reason for this dilemma is of a conceptual nature: the lack of clarity about the essential questions. The paper contrasts the usually applied simulation methods with the minimal error simulation methods presented recently. The comparisons are used to address essential questions about the required characteristics of the desired simulation methods. The advantages of novel simulation methods (including their simplicity, significant computational cost reductions, and controlled resolution ability) are pointed out. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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