Research

9 pages, 746 KiB

Open AccessArticle

Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions

by Saad Althobati, Jehad Alzabut and Omar Bazighifan

Symmetry 2021, 13(2), 285; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020285 - 07 Feb 2021

Cited by 6 | Viewed by 1309

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

17 pages, 1409 KiB

Open AccessArticle

Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

by Yasser Salah Hamed and Ali Kandil

Symmetry 2021, 13(1), 85; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13010085 - 06 Jan 2021

Cited by 7 | Viewed by 1322

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

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11 pages, 98992 KiB

Open AccessArticle

Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions

by Sameh S. Askar and Abdulrahman Al-Khedhairi

Symmetry 2020, 12(12), 2001; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12122001 - 04 Dec 2020

Cited by 2 | Viewed by 1493

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

Z_{0}, Z_{2}

and

Z_{4}

regions. Full article

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

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23 pages, 6984 KiB

Open AccessArticle

Utilizing Macro Fiber Composite to Control Rotating Blade Vibrations

by Y. S. Hamed, Ali Kandil and José Tenreiro Machado

Symmetry 2020, 12(12), 1984; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12121984 - 30 Nov 2020

Cited by 10 | Viewed by 1464

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

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12 pages, 761 KiB

Open AccessArticle

Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique

by Mutti-Ur Rehman, Jehad Alzabut and Kamaleldin Abodayeh

Symmetry 2020, 12(11), 1824; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111824 - 04 Nov 2020

Cited by 1 | Viewed by 1345

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

10 pages, 241 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Some New Oscillation Results for Fourth-Order Neutral Differential Equations with Delay Argument

by Omar Bazighifan, Osama Moaaz, Rami Ahmad El-Nabulsi and Ali Muhib

Symmetry 2020, 12(8), 1248; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12081248 - 29 Jul 2020

Cited by 15 | Viewed by 2165

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

9 pages, 237 KiB

Open AccessArticle

Oscillation Conditions for Certain Fourth-Order Non-Linear Neutral Differential Equation

by Ioannis Dassios and Omar Bazighifan

Symmetry 2020, 12(7), 1096; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12071096 - 02 Jul 2020

Cited by 15 | Viewed by 1732

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

26 pages, 447 KiB

Open AccessArticle

Discrete-Time Stochastic Quaternion-Valued Neural Networks with Time Delays: An Asymptotic Stability Analysis

by Ramalingam Sriraman, Grienggrai Rajchakit, Chee Peng Lim, Pharunyou Chanthorn and Rajendran Samidurai

Symmetry 2020, 12(6), 936; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12060936 - 03 Jun 2020

Cited by 41 | Viewed by 2296

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

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11 pages, 770 KiB

Open AccessArticle

New Comparison Theorems for the Even-Order Neutral Delay Differential Equation

by Osama Moaaz, Rami Ahmad El-Nabulsi, Omar Bazighifan and Ali Muhib

Symmetry 2020, 12(5), 764; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12050764 - 06 May 2020

Cited by 6 | Viewed by 1438

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

12 pages, 258 KiB

Open AccessFeature PaperArticle

Behavior of Non-Oscillatory Solutions of Fourth-Order Neutral Differential Equations

by Osama Moaaz, Rami Ahmad El-Nabulsi and Omar Bazighifan

Symmetry 2020, 12(3), 477; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12030477 - 19 Mar 2020

Cited by 9 | Viewed by 1735

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

(This article belongs to the Special Issue Asymptotic Properties of Solutions of Difference and Differential Equations)

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