Special Issue "Asymptotic Properties of Solutions of Difference and Differential Equations"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics and Symmetry/Asymmetry".

Deadline for manuscript submissions: closed (30 December 2020).

Special Issue Editors

Prof. Dr. Rami Ahmad El-Nabulsi
E-Mail Website
Guest Editor
1. Athens Institute for Education and Research, Mathematics and Physics Divisions, 10671 Athens, Greece
2. Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3. Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Interests: geometrical dynamics; quantum mechanics; nonlinearity; fractal dynamics; geometrical physics; general relativity and gravitation; operators theory; quantum field theory; plasma MHD and planetary dynamics; chaos and bifurcations; reactor physics and nuclear sciences; solid state physics and magnetism; quantum electronics and nanostructures
Special Issues, Collections and Topics in MDPI journals
Dr. Osama Moaaz
E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Interests: differental equations; numerical analysis; analysis; applied mathematics; nonlinear dynamics; mathematical modelling; mathematical analysis; stability
Special Issues, Collections and Topics in MDPI journals
Dr. Omar Bazighifan
E-Mail Website
Guest Editor
1. Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
2. Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen
Interests: qualitative theory; ordinary differential equations; functional differential equations; dynamical systems; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

Difference and differential equations have been used since Newton’s time for the understanding of physical sciences, engineering, and vitality, as well as for sport, economic, and social sciences. This is because most of the relationships between variables and laws governing both physical and engineering issues and natural phenomena can be represented by differential equations. By solving these equations, it is possible to describe and understand these issues and phenomena. However, differential equations, such as those used to solve real-life problems, may not be directly solvable, i.e., they do not have closed-form solutions. Only the simplest equations admit solutions obtained from explicit formulas. Despite this, some properties of the solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for a solution is not available, the solution may be numerically approximated using computers. In this case, a recurrence relation is needed. This is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.

In recent decades, various models of difference and differential equations have been proposed in different sciences, strongly motivating research in the qualitative theory of difference and differential equations. Symmetry ideas are often missing in these studies, but they help us find the most suitable approach to studying difference and differential equations and suggest correct directions for future developments. The twofold purpose of this Special Issue is to bring together state-of-the-art theoretical research and its applications in mathematical models.

Prof. Rami Ahmad El-Nabulsi
Dr. Osama Moaaz
Dr. Omar Bazighifan
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 papers will be 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. Symmetry 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 1800 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

  • ordinary differential equation
  • difference equation
  • partial differential equation
  • symmetry numerical analysis
  • approximation, stability, oscillation, boundedness, periodicity, asymptotic properties

Published Papers (10 papers)

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Research

Article
Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions
Symmetry 2021, 13(2), 285; https://doi.org/10.3390/sym13020285 - 07 Feb 2021
Cited by 5 | Viewed by 477
Abstract
The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for [...] Read more.
The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for a class of nonlinear even-order differential equations with damping. We employ different approach based on using Riccati technique to reduce the main equation into a second order equation and then comparing with a second order equation whose oscillatory behavior is known. The new conditions complement several results in the literature. Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples. Full article
Article
Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade
Symmetry 2021, 13(1), 85; https://doi.org/10.3390/sym13010085 - 06 Jan 2021
Cited by 3 | Viewed by 436
Abstract
Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors [...] Read more.
Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors on the blade, and its output is the suitable control force applied onto the actuators on the blade driving it to the desired minimum vibratory level. Based on the saturation phenomenon, the blade vibrations can be saturated at a specific level while the rest of the energy is transferred to the controller. This can be done by adjusting the controller natural frequency to be one half of the blade natural frequency. The whole behavior is governed by a system of first-order differential equations gained by the method of multiple scales. Different responses are included to show the influences of time delay on the closed-loop control process. Also, a good agreement can be noticed between the analytical curves and the numerically simulated ones. Full article
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Article
Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions
Symmetry 2020, 12(12), 2001; https://doi.org/10.3390/sym12122001 - 04 Dec 2020
Viewed by 411
Abstract
In this paper, we study the complex dynamic characteristics of a simple nonlinear logistic map. The map contains two parameters that have complex influences on the map’s dynamics. Assuming different values for those parameters gives rise to strange attractors with fractal dimensions. Furthermore, [...] Read more.
In this paper, we study the complex dynamic characteristics of a simple nonlinear logistic map. The map contains two parameters that have complex influences on the map’s dynamics. Assuming different values for those parameters gives rise to strange attractors with fractal dimensions. Furthermore, some of these chaotic attractors have heteroclinic cycles due to saddle-fixed points. The basins of attraction for some periodic cycles in the phase plane are divided into three regions of rank-1 preimages. We analyze those regions and show that the map is noninvertible and includes Z0,Z2 and Z4 regions. Full article
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Article
Utilizing Macro Fiber Composite to Control Rotating Blade Vibrations
Symmetry 2020, 12(12), 1984; https://doi.org/10.3390/sym12121984 - 30 Nov 2020
Cited by 5 | Viewed by 526
Abstract
This work applies an active control algorithm, using a macro fiber composite (MFC) to mitigate the unwanted vibrations of a rotating blade. The algorithm is a second-order oscillator, having the positive displacement signal of the blade for input and the suitable control force [...] Read more.
This work applies an active control algorithm, using a macro fiber composite (MFC) to mitigate the unwanted vibrations of a rotating blade. The algorithm is a second-order oscillator, having the positive displacement signal of the blade for input and the suitable control force to actuate the blade for output. This oscillator is linearly coupled with the blade, having in mind that their natural frequencies must be in the vicinity of each other. The rotating blade is modeled by representing two vibrational directions that are linearly coupled. An asymptotic analysis is considered to understand the resulting nonlinear phenomena. Several responses are included to portray the dynamical behavior of the system under control. From the results, we observe the asymmetry between the blade’s vibrational directions. Moreover, a verification is included for comparing the analytical and numerical results. Full article
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Article
Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique
Symmetry 2020, 12(11), 1824; https://doi.org/10.3390/sym12111824 - 04 Nov 2020
Cited by 1 | Viewed by 487
Abstract
For n-dimensional real-valued matrix A, the computation of nearest correlation matrix; that is, a symmetric, positive semi-definite, unit diagonal and off-diagonal entries between 1 and 1 is a problem that arises in the finance industry where the correlations exist between [...] Read more.
For n-dimensional real-valued matrix A, the computation of nearest correlation matrix; that is, a symmetric, positive semi-definite, unit diagonal and off-diagonal entries between 1 and 1 is a problem that arises in the finance industry where the correlations exist between the stocks. The proposed methodology presented in this article computes the admissible perturbation matrix and a perturbation level to shift the negative spectrum of perturbed matrix to become non-negative or strictly positive. The solution to optimization problems constructs a gradient system of ordinary differential equations that turn over the desired perturbation matrix. Numerical testing provides enough evidence for the shifting of the negative spectrum and the computation of nearest correlation matrix. Full article
Article
Some New Oscillation Results for Fourth-Order Neutral Differential Equations with Delay Argument
Symmetry 2020, 12(8), 1248; https://doi.org/10.3390/sym12081248 - 29 Jul 2020
Cited by 8 | Viewed by 736
Abstract
The aim of this paper is to study the oscillatory properties of 4th-order neutral differential equations. We obtain some oscillation criteria for the equation by the theory of comparison. The obtained results improve well-known oscillation results in the literate. Symmetry plays an important [...] Read more.
The aim of this paper is to study the oscillatory properties of 4th-order neutral differential equations. We obtain some oscillation criteria for the equation by the theory of comparison. The obtained results improve well-known oscillation results in the literate. Symmetry plays an important role in determining the right way to study these equation. An example to illustrate the results is given. Full article
Article
Oscillation Conditions for Certain Fourth-Order Non-Linear Neutral Differential Equation
Symmetry 2020, 12(7), 1096; https://doi.org/10.3390/sym12071096 - 02 Jul 2020
Cited by 4 | Viewed by 586
Abstract
In this work, new conditions were obtained for the oscillation of solutions of fourth-order non-linear neutral differential equations (NDEs) using the Riccati technique. These oscillation criteria complement and improve those of Chatzarakis et al. (2019). Symmetry plays an important role in determining the [...] Read more.
In this work, new conditions were obtained for the oscillation of solutions of fourth-order non-linear neutral differential equations (NDEs) using the Riccati technique. These oscillation criteria complement and improve those of Chatzarakis et al. (2019). Symmetry plays an important role in determining the right way to study these equation. An example is given to illustrate our theory. Full article
Article
Discrete-Time Stochastic Quaternion-Valued Neural Networks with Time Delays: An Asymptotic Stability Analysis
Symmetry 2020, 12(6), 936; https://doi.org/10.3390/sym12060936 - 03 Jun 2020
Cited by 19 | Viewed by 860
Abstract
Stochastic disturbances often cause undesirable characteristics in real-world system modeling. As a result, investigations on stochastic disturbances in neural network (NN) modeling are important. In this study, stochastic disturbances are considered for the formulation of a new class of NN models; i.e., the [...] Read more.
Stochastic disturbances often cause undesirable characteristics in real-world system modeling. As a result, investigations on stochastic disturbances in neural network (NN) modeling are important. In this study, stochastic disturbances are considered for the formulation of a new class of NN models; i.e., the discrete-time stochastic quaternion-valued neural networks (DSQVNNs). In addition, the mean-square asymptotic stability issue in DSQVNNs is studied. Firstly, we decompose the original DSQVNN model into four real-valued models using the real-imaginary separation method, in order to avoid difficulties caused by non-commutative quaternion multiplication. Secondly, some new sufficient conditions for the mean-square asymptotic stability criterion with respect to the considered DSQVNN model are obtained via the linear matrix inequality (LMI) approach, based on the Lyapunov functional and stochastic analysis. Finally, examples are presented to ascertain the usefulness of the obtained theoretical results. Full article
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Article
New Comparison Theorems for the Even-Order Neutral Delay Differential Equation
Symmetry 2020, 12(5), 764; https://doi.org/10.3390/sym12050764 - 06 May 2020
Cited by 3 | Viewed by 583
Abstract
The aim of this study was to examine the asymptotic properties and oscillation of the even-order neutral differential equations. The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order delay equations. Our results improve and [...] Read more.
The aim of this study was to examine the asymptotic properties and oscillation of the even-order neutral differential equations. The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order delay equations. Our results improve and complement some well-known results. We obtain Hille and Nehari type oscillation criteria to ensure the oscillation of the solutions of the equation. One example is provided to illustrate these results. Full article
Article
Behavior of Non-Oscillatory Solutions of Fourth-Order Neutral Differential Equations
Symmetry 2020, 12(3), 477; https://doi.org/10.3390/sym12030477 - 19 Mar 2020
Cited by 4 | Viewed by 753
Abstract
In this paper, we deal with the asymptotics and oscillation of the solutions of fourth-order neutral differential equations of the form r t z t α + q t x α g t = 0 , where [...] Read more.
In this paper, we deal with the asymptotics and oscillation of the solutions of fourth-order neutral differential equations of the form r t z t α + q t x α g t = 0 , where z t : = x t + p t x δ t . By using a generalized Riccati transformation, we study asymptotic behavior and derive some new oscillation criteria. Our results extend and improve some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results. Full article
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