Mathematics

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9 pages, 263 KiB

Open AccessArticle

New Criteria of Oscillation for Linear Sturm–Liouville Delay Noncanonical Dynamic Equations

by Taher S. Hassan, Martin Bohner, Iambor Loredana Florentina, Amir Abdel Menaem and Mouataz Billah Mesmouli

Mathematics 2023, 11(23), 4850; https://doi.org/10.3390/math11234850 - 01 Dec 2023

Cited by 2 | Viewed by 719

Abstract

In this work, we deduce a new criterion that guarantees the oscillation of solutions to linear Sturm–Liouville delay noncanonical dynamic equations; these results emulate the criteria of the Hille and Ohriska types for canonical dynamic equations, and these results also solve an open [...] Read more.

In this work, we deduce a new criterion that guarantees the oscillation of solutions to linear Sturm–Liouville delay noncanonical dynamic equations; these results emulate the criteria of the Hille and Ohriska types for canonical dynamic equations, and these results also solve an open problem in many works in the literature. Several examples are offered, demonstrating that the findings achieved are precise, practical, and adaptable. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

16 pages, 7770 KiB

Open AccessCommunication

Chaotic Vibration and Perforation Effects on the Sound Absorption of a Nonlinear Curved Panel Absorber

by Yiu-Yin Lee

Mathematics 2023, 11(14), 3178; https://doi.org/10.3390/math11143178 - 20 Jul 2023

Cited by 1 | Viewed by 611

Abstract

This study is the first to investigate the effect of chaotic vibration on the sound absorption of a curved perforated panel. Previous studies on the effect of nonlinear vibration on the sound absorption of a panel absorber have focused on periodic responses only. [...] Read more.

This study is the first to investigate the effect of chaotic vibration on the sound absorption of a curved perforated panel. Previous studies on the effect of nonlinear vibration on the sound absorption of a panel absorber have focused on periodic responses only. In this study, a sound absorption formula was derived by considering the panel impedance and perforation impedance. The numerical integration method was adopted to generate various chaotic vibrational responses, which were used to compute the corresponding sound absorptions. Several interesting findings that have never been observed in any previous studies on acoustic absorption were derived. First, in the chaotic and highly nonlinear cases, as the excitation frequency increased, the corresponding response frequencies decreased. This was opposite to the typical trend in linear cases, in which higher excitation frequencies corresponded to higher response frequencies. Second, in chaotic cases, absorption mainly occurred due to panel vibration effects. This is also in stark contrast to the findings of studies on perforated vibrating panels, in which the absorption effect mainly originates from perforations. Additionally, the absorption bandwidths are much wider and can shift to higher frequencies; however, the peak absorption coefficients were approximately 20% lower than in the case of the perforation effect only. Third, in the quasi-chaotic case, the absorption curve in the case of the perforation effect plus the vibration effect was between the absorption curves of the perforation effect only and the perforation effect plus the vibration effect. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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13 pages, 13982 KiB

Open AccessArticle

Chaotic States of Transistor-Based Tuned-Collector Oscillator

by Jiri Petrzela

Mathematics 2023, 11(9), 2213; https://doi.org/10.3390/math11092213 - 08 May 2023

Cited by 2 | Viewed by 876

Abstract

This brief paper shows that robust chaotic behavior can be detected within a tuned-collector single-stage transistor-based oscillator. The content of this work also contributes to the problem of chaos localization in simplified mathematical model of standard analog building block. Searching for chaos is [...] Read more.

This brief paper shows that robust chaotic behavior can be detected within a tuned-collector single-stage transistor-based oscillator. The content of this work also contributes to the problem of chaos localization in simplified mathematical model of standard analog building block. Searching for chaos is performed via numerical optimization routine applied onto the principal schematic of oscillator where generalized bipolar transistor is modelled as a two-port described by impedance as well as admittance matrix. In both cases, the presence of dense chaotic attractor is proved via calculation of the largest Lyapunov exponent, while its structural stability is validated by real measurement, i.e., visualization of captured oscilloscope screenshots. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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9 pages, 4541 KiB

Open AccessArticle

Canards Oscillations, Noise-Induced Splitting of Cycles and Transition to Chaos in Thermochemical Kinetics

by Irina Bashkirtseva, Grigoriy Ivanenko, Dmitrii Mordovskikh and Lev Ryashko

Mathematics 2023, 11(8), 1918; https://doi.org/10.3390/math11081918 - 19 Apr 2023

Cited by 1 | Viewed by 715

Abstract

We study how noise generates complex oscillatory regimes in the nonlinear thermochemical kinetics. In this study, the basic mathematical Zeldovich–Semenov model is used as a deterministic skeleton. We investigate the stochastic version of this model that takes into account multiplicative random fluctuations of [...] Read more.

We study how noise generates complex oscillatory regimes in the nonlinear thermochemical kinetics. In this study, the basic mathematical Zeldovich–Semenov model is used as a deterministic skeleton. We investigate the stochastic version of this model that takes into account multiplicative random fluctuations of temperature. In our study, we use direct numerical simulation of stochastic solutions with the subsequent statistical analysis of probability densities and Lyapunov exponents. In the parametric zone of Canard cycles, qualitative effects caused by random noise are identified and investigated. Stochastic P-bifurcations corresponding to noise-induced splitting of Canard oscillations are parametrically described. It is shown that such P-bifurcations are associated with splitting of both amplitudes and frequencies. Studying stochastic D-bifurcations, we localized the rather narrow parameter zone where transitions from order to chaos occur. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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10 pages, 279 KiB

Open AccessArticle

Oscillation Criteria for Advanced Half-Linear Differential Equations of Second Order

by Taher S. Hassan, Qingkai Kong and Bassant M. El-Matary

Mathematics 2023, 11(6), 1385; https://doi.org/10.3390/math11061385 - 13 Mar 2023

Cited by 4 | Viewed by 963

Abstract

In this paper, we find new oscillation criteria for second-order advanced functional half-linear differential equations. Our results extend and improve recent criteria for the same equations established previously by several authors and cover the existing classical criteria for related ordinary differential equations. We [...] Read more.

In this paper, we find new oscillation criteria for second-order advanced functional half-linear differential equations. Our results extend and improve recent criteria for the same equations established previously by several authors and cover the existing classical criteria for related ordinary differential equations. We give some examples to illustrate the significance of the obtained results. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

10 pages, 273 KiB

Open AccessArticle

Asymptotic Behavior and Oscillation of Third-Order Nonlinear Neutral Differential Equations with Mixed Nonlinearities

by Taher S. Hassan and Bassant M. El-Matary

Mathematics 2023, 11(2), 424; https://doi.org/10.3390/math11020424 - 13 Jan 2023

Cited by 1 | Viewed by 880

Abstract

In this paper, we investigate the asymptotic properties of third-order nonlinear neutral differential equations with mixed nonlinearities using the comparison principle. Our results not only vastly improve upon but also broadly generalize many previously known ones. Examples demonstrating the applicability and efficacy of [...] Read more.

In this paper, we investigate the asymptotic properties of third-order nonlinear neutral differential equations with mixed nonlinearities using the comparison principle. Our results not only vastly improve upon but also broadly generalize many previously known ones. Examples demonstrating the applicability and efficacy of our results are provided. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

14 pages, 453 KiB

Open AccessArticle

New Explicit Oscillation Criteria for First-Order Differential Equations with Several Non-Monotone Delays

by Emad Attia and Bassant El-Matary

Mathematics 2023, 11(1), 64; https://doi.org/10.3390/math11010064 - 24 Dec 2022

Cited by 1 | Viewed by 1032

Abstract

The oscillation of a first-order differential equation with several non-monotone delays is proposed. We extend the works of Kwong (1991) and Sficas and Stavroulakis (2003) for equations with several delays. Our results not only essentially improve but also generalize a large number of [...] Read more.

The oscillation of a first-order differential equation with several non-monotone delays is proposed. We extend the works of Kwong (1991) and Sficas and Stavroulakis (2003) for equations with several delays. Our results not only essentially improve but also generalize a large number of the existing ones. Using some numerical examples, we illustrate the applicability and effectiveness of our results over many known results in the literature. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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13 pages, 723 KiB

Open AccessArticle

Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

by Alvaro H. Salas, Ma’mon Abu Hammad, Badriah M. Alotaibi, Lamiaa S. El-Sherif and Samir A. El-Tantawy

Mathematics 2022, 10(21), 4000; https://doi.org/10.3390/math10214000 - 28 Oct 2022

Cited by 3 | Viewed by 1118

Abstract

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for [...] Read more.

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for the conserved case is obtained in the form of a trigonometric function. A comparison between both exact and approximate solutions to the conserved oscillator is examined. Moreover, the analytical approximations to the non-conserved oscillators including the unforced, damped rotational pendulum oscillator and forced, damped rotational pendulum oscillator are obtained. Furthermore, all mentioned oscillators (conserved and non-conserved oscillators) are linearized, and their exact solutions are derived. In addition, all obtained approximations are compared with the four-order Runge–Kutta (RK4) numerical approximations and with the exact solutions to the linearized oscillators. The obtained results can help several authors for discussing and interpreting their results. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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18 pages, 326 KiB

Open AccessArticle

Improved Hille-Type Oscillation Criteria for Second-Order Quasilinear Dynamic Equations

by Taher S. Hassan, Clemente Cesarano, Rami Ahmad El-Nabulsi and Waranont Anukool

Mathematics 2022, 10(19), 3675; https://doi.org/10.3390/math10193675 - 07 Oct 2022

Cited by 5 | Viewed by 897

Abstract

In this work, we develop enhanced Hille-type oscillation conditions for arbitrary-time, second-order quasilinear functional dynamic equations. These findings extend and improve previous research that has been published in the literature. Some examples are given to demonstrate the importance of the obtained results. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

19 pages, 1051 KiB

Open AccessArticle

Time-Optimal Control Problem of Two Non-Synchronous Oscillators

by Leonid Berlin, Andrey Galyaev and Pavel Lysenko

Mathematics 2022, 10(19), 3552; https://doi.org/10.3390/math10193552 - 29 Sep 2022

Cited by 2 | Viewed by 1064

Abstract

The time-optimal control problem for a system consisting of two non-synchronous oscillators is considered. Each oscillator is controlled with a shared limited scalar control. The objective of the control is to accelerate the oscillatory system to a given specific position, where the first [...] Read more.

The time-optimal control problem for a system consisting of two non-synchronous oscillators is considered. Each oscillator is controlled with a shared limited scalar control. The objective of the control is to accelerate the oscillatory system to a given specific position, where the first oscillator must have non-zero phase coordinates, but the second one must remain motionless at the terminal moment. For an arbitrary number of unknown switching moments that determine the optimal relay control, the necessary extremum conditions in the form of nonlinear matrix equalities are proposed. The study of the necessary/sufficient conditions of the extremum made it possible to describe the reachability set in the phase space of the first oscillator, to find an analytical form of the curve corresponding to the two-switching control class, which also separates the reachability set of the three switching-control class. The corresponding theorems are proved and the dependence of the criteria on control constraints is shown. Matrix conditions for different classes of control switchings are found. All of the obtained analytical results are numerically validated and illustrated with mathematical modeling. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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16 pages, 781 KiB

Open AccessArticle

Numerical Solutions of Fractional-Order Electrical RLC Circuit Equations via Three Numerical Techniques

by Uroosa Arshad, Mariam Sultana, Ali Hasan Ali, Omar Bazighifan, Areej A. Al-moneef and Kamsing Nonlaopon

Mathematics 2022, 10(17), 3071; https://doi.org/10.3390/math10173071 - 25 Aug 2022

Cited by 14 | Viewed by 2180

Abstract

In this article, three different techniques, the Fractional Perturbation Iteration Method (FPIA), Fractional Successive Differentiation Method (FSDM), and Fractional Novel Analytical Method (FNAM), have been introduced. These three iterative methods are applied on different types of Electrical RLC-Circuit Equations of fractional-order. The fractional [...] Read more.

In this article, three different techniques, the Fractional Perturbation Iteration Method (FPIA), Fractional Successive Differentiation Method (FSDM), and Fractional Novel Analytical Method (FNAM), have been introduced. These three iterative methods are applied on different types of Electrical RLC-Circuit Equations of fractional-order. The fractional series approximation of the derived solutions can be established by using the obtained coefficients. These three algorithms handle the problems in a direct manner without any need for restrictive assumptions. The comparison displays an agreement between the obtained results. The beauty of this paper lies in the error analysis between the exact solution and approximate solutions obtained by these three methods which prove that the Approximate Solution obtained by FNAM converge very rapidly to the exact solution. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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11 pages, 284 KiB

Open AccessArticle

Some New Oscillation Results for Higher-Order Nonlinear Differential Equations with a Nonlinear Neutral Term

by John R. Graef, Said R. Grace, Irena Jadlovská and Ercan Tunç

Mathematics 2022, 10(16), 2997; https://doi.org/10.3390/math10162997 - 19 Aug 2022

Cited by 1 | Viewed by 985

Abstract

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or [...] Read more.

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or superlinear. The influence of these terms on the oscillatory and asymptotic behaviors of solutions is investigated by using a comparison to first-order advanced and delay differential equations. New oscillation criteria are presented that improve and extend many known oscillation criteria in the literature. An example is provided to illustrate the results. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

11 pages, 256 KiB

Open AccessArticle

Limit Cycles and Integrability of a Class of Quintic System

by Yanli Tang, Dongmei Zhang and Feng Li

Mathematics 2022, 10(16), 2993; https://doi.org/10.3390/math10162993 - 19 Aug 2022

Viewed by 857

Abstract

In this paper, a class of quintic systems is investigated. The first 13 focal values are computed with the aid of MATHEMATICA. Then the necessary conditions of integrability and linearizability are obtained and the sufficiency of every condition is proved. Meanwhile, bifurcation of [...] Read more.

In this paper, a class of quintic systems is investigated. The first 13 focal values are computed with the aid of MATHEMATICA. Then the necessary conditions of integrability and linearizability are obtained and the sufficiency of every condition is proved. Meanwhile, bifurcation of limit cycles is discussed, 13 limit cycles can be bifurcated from the origin. As far as the number of limit cycles enclosing an isolated singular point is concerned, this is so far the best result for elementary singular points. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

16 pages, 3269 KiB

Open AccessArticle

Detection and Analysis of Critical Dynamic Properties of Oligodendrocyte Differentiation

by Svetoslav G. Nikolov, Olaf Wolkenhauer, Momchil Nenov and Julio Vera

Mathematics 2022, 10(16), 2928; https://doi.org/10.3390/math10162928 - 14 Aug 2022

Cited by 1 | Viewed by 1080

Abstract

In this paper, we derive a four-dimensional ordinary differential equation (ODE) model representing the main interactions between Sox9, Sox10, Olig2 and several miRNAs, which drive the process of (olygodendrocyte) differentiation. We utilize the Lyapunov–Andronov theory to analyze its dynamical properties. Our results indicated [...] Read more.

In this paper, we derive a four-dimensional ordinary differential equation (ODE) model representing the main interactions between Sox9, Sox10, Olig2 and several miRNAs, which drive the process of (olygodendrocyte) differentiation. We utilize the Lyapunov–Andronov theory to analyze its dynamical properties. Our results indicated that the strength of external signaling (morphogenic gradients shh and bmp), and the transcription rate of mOlig2 explain the existence of stable and unstable (sustained oscillations) behavior in the system. Possible biological implications are discussed. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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12 pages, 775 KiB

Open AccessArticle

Novel Approximations to the (Un)forced Pendulum–Cart System: Ansatz and KBM Methods

by Weaam Alhejaili, Alvaro H. Salas and Samir A. El-Tantawy

Mathematics 2022, 10(16), 2908; https://doi.org/10.3390/math10162908 - 12 Aug 2022

Cited by 10 | Viewed by 1015

Abstract

In the present investigation, some novel analytical approximations to both unforced and forced pendulum–cart system oscillators are obtained. In our investigation, two accurate and effective approaches, namely, the ansatz method with equilibrium point and the Krylov–Bogoliubov–Mitropolsky (KBM) method, are implemented for analyzing pendulum–cart [...] Read more.

In the present investigation, some novel analytical approximations to both unforced and forced pendulum–cart system oscillators are obtained. In our investigation, two accurate and effective approaches, namely, the ansatz method with equilibrium point and the Krylov–Bogoliubov–Mitropolsky (KBM) method, are implemented for analyzing pendulum–cart problems.The obtained results are compared with the Runge–Kutta (RK4) numerical approximation. The obtained approximations using both ansatz and KBM methods show good coincidence with RK4 numerical approximation. In addition, the global maximum error is estimated as compared to RK4 numerical approximation. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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12 pages, 292 KiB

Open AccessArticle

On the Asymptotic Behavior of Noncanonical Third-Order Emden–Fowler Delay Differential Equations with a Superlinear Neutral Term

by Qingmin Liu, Said R. Grace, Irena Jadlovská, Ercan Tunç and Tongxing Li

Mathematics 2022, 10(16), 2902; https://doi.org/10.3390/math10162902 - 12 Aug 2022

Cited by 2 | Viewed by 912

Abstract

The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that [...] Read more.

The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that every solution of the equation considered either oscillates or converges to zero asymptotically. The results are illustrated with two examples. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

12 pages, 280 KiB

Open AccessArticle

Oscillation of Second Order Nonlinear Neutral Differential Equations

by Yingzhu Wu, Yuanhong Yu and Jinsen Xiao

Mathematics 2022, 10(15), 2739; https://doi.org/10.3390/math10152739 - 02 Aug 2022

Cited by 2 | Viewed by 989

Abstract

The study of the oscillatory behavior of solutions to second order nonlinear differential equations is motivated by their numerous applications in the natural sciences and engineering. In the presented research, some new oscillation criteria for a class of damped second order neutral differential [...] Read more.

The study of the oscillatory behavior of solutions to second order nonlinear differential equations is motivated by their numerous applications in the natural sciences and engineering. In the presented research, some new oscillation criteria for a class of damped second order neutral differential equations with noncanonical operators are established. The results extend and improve on those reported in the literature. Moreover, some examples are provided to show the significance of the results. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)

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