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14 pages, 336 KiB

Open AccessArticle

Systematic Review of Geometrical Approaches and Analytical Integration for Chen’s System

by Remus-Daniel Ene, Camelia Pop and Camelia Petrişor

Mathematics 2020, 8(9), 1530; https://doi.org/10.3390/math8091530 - 8 Sep 2020

Cited by 1 | Viewed by 1508

The main goal of this paper is to present an analytical integration in connection with the geometrical frame given by the Hamilton–Poisson formulation of a specific case of Chen’s system. In this special case we construct an analytic approximate solution using the Multistage [...] Read more.

The main goal of this paper is to present an analytical integration in connection with the geometrical frame given by the Hamilton–Poisson formulation of a specific case of Chen’s system. In this special case we construct an analytic approximate solution using the Multistage Optimal Homotopy Asymptotic Method (MOHAM). Numerical simulations are also presented in order to make a comparison between the analytic approximate solution and the corresponding numerical solution. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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13 pages, 1727 KiB

Open AccessArticle

Vibration of the Biomass Boiler Tube Excited with Impact of the Cleaning Device

by Dragan Cveticanin, Nicolae Herisanu, Istvan Biro, Miodrag Zukovic and Livija Cveticanin

Mathematics 2020, 8(9), 1519; https://doi.org/10.3390/math8091519 - 6 Sep 2020

Cited by 3 | Viewed by 2431

Abstract

In boilers with biomass fuel, a significant problem is caused due to the slag layer formed from the unburned particles during combustion. In the paper, a tube cleaning method from slag is proposed. The method is based on the impact effect of the [...] Read more.

In boilers with biomass fuel, a significant problem is caused due to the slag layer formed from the unburned particles during combustion. In the paper, a tube cleaning method from slag is proposed. The method is based on the impact effect of the end of the tube with the aim to produce vibration for slag elimination. The tube is modeled as a clamped-free nonlinear oscillatory system. The initial impact of the tube causes vibrations. The mathematical model of the system is a nonlinear partial differential equation with zero initial deflection. To obtain the ordinary differential equations, the Galerkin method is applied. By discretizing the equation into a finite degree of freedom system, using the undamped linear mode shapes of the straight beam as basic functions, the reduced order model, consisting of ordinary differential equations in time, is obtained. The ordinary time equations are analytically solved by adopting the Krylov–Bogoliubov procedure. Special cases of nonlinear differential equations are investigated. In the paper, the influence of the nonlinear parameters and initial conditions on the vibration properties of the tube is obtained. We use the procedure developed in the paper and the analytical results for computation of the impact parameters of the cleaning device. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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15 pages, 1048 KiB

Open AccessArticle

Numerical Simulation of Flow over Non-Linearly Stretching Sheet Considering Chemical Reaction and Magnetic Field

by Mohsen Razzaghi, Fatemeh Baharifard and Kourosh Parand

Mathematics 2020, 8(9), 1496; https://doi.org/10.3390/math8091496 - 4 Sep 2020

Cited by 1 | Viewed by 2001

Abstract

The purpose of this paper is to investigate a system of differential equations related to the viscous flow over a stretching sheet. It is assumed that the intended environment for the flow includes a chemical reaction and a magnetic field. The governing equations [...] Read more.

The purpose of this paper is to investigate a system of differential equations related to the viscous flow over a stretching sheet. It is assumed that the intended environment for the flow includes a chemical reaction and a magnetic field. The governing equations are defined on the semi-finite domain and a numerical scheme, namely rational Gegenbauer collocation method is applied to solve it. In this method, the problem is solved in its main interval (semi-infinite domain) and there is no need to truncate it to a finite domain or change the domain of the problem. By carefully examining the effect of important physical parameters of the problem and comparing the obtained results with the answers of other methods, we show that despite the simplicity of the proposed method, it has a high degree of convergence and good accuracy. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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18 pages, 4237 KiB

Open AccessArticle

Optimal Auxiliary Functions Method for a Pendulum Wrapping on Two Cylinders

by Vasile Marinca and Nicolae Herisanu

Mathematics 2020, 8(8), 1364; https://doi.org/10.3390/math8081364 - 14 Aug 2020

Cited by 18 | Viewed by 3088

Abstract

In the present work, the nonlinear oscillations of a pendulum wrapping on two cylinders is studied by means of a new analytical technique, namely the Optimal Auxiliary Functions Method (OAFM). The equation of motion is derived from the Lagrange’s equation. Analytical solutions and [...] Read more.

In the present work, the nonlinear oscillations of a pendulum wrapping on two cylinders is studied by means of a new analytical technique, namely the Optimal Auxiliary Functions Method (OAFM). The equation of motion is derived from the Lagrange’s equation. Analytical solutions and natural frequency of the system are calculated. Our results obtained through this new procedure are compared with numerical ones and a very good agreement was found, which proves the accuracy of the method. The presented numerical examples show that the proposed approach is simple, easy to implement and very accurate. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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12 pages, 366 KiB

Open AccessArticle

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

by Constantin Bota, Bogdan Căruntu, Dumitru Ţucu, Marioara Lăpădat and Mădălina Sofia Paşca

Mathematics 2020, 8(8), 1336; https://doi.org/10.3390/math8081336 - 11 Aug 2020

Cited by 9 | Viewed by 2198

Abstract

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the [...] Read more.

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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16 pages, 2464 KiB

Open AccessArticle

Study of Lotka–Volterra Biological or Chemical Oscillator Problem Using the Normalization Technique: Prediction of Time and Concentrations

by Juan Francisco Sánchez-Pérez, Manuel Conesa, Iván Alhama and Manuel Cánovas

Mathematics 2020, 8(8), 1324; https://doi.org/10.3390/math8081324 - 9 Aug 2020

Cited by 13 | Viewed by 2420

Abstract

The normalization of dimensionless groups that rule the system of nonlinear coupled ordinary differential equations defined by the Lotka–Volterra biological or chemical oscillator has been derived in this work by applying a normalized nondimensionalization protocol. The normalization procedure, which is quite accurate, does [...] Read more.

The normalization of dimensionless groups that rule the system of nonlinear coupled ordinary differential equations defined by the Lotka–Volterra biological or chemical oscillator has been derived in this work by applying a normalized nondimensionalization protocol. The normalization procedure, which is quite accurate, does not require complex mathematical steps; however, a deep physical understanding of the problem is required to choose the appropriate references to define the dimensionless variables. From the dimensionless groups derived, the functional dependences of some unknowns of interest are established. Due to the coupled nature of the problem that induces temporal concentration rates of each species that are quite different at each point of the phase diagram, this diagram has been divided into four stretches corresponding to the four quadrants. For each stretch, the limit values (maximum or minimum) of the variables, as well as their duration, are expressed in terms of the dimensionless groups derived before. Finally, to check all the mentioned dependences, a numerical simulation has been carried out. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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20 pages, 417 KiB

Open AccessArticle

Entrainment of Weakly Coupled Canonical Oscillators with Applications in Gradient Frequency Neural Networks Using Approximating Analytical Methods

by AmirAli Farokhniaee, Felix V. Almonte, Susanne Yelin and Edward W. Large

Mathematics 2020, 8(8), 1312; https://doi.org/10.3390/math8081312 - 7 Aug 2020

Cited by 3 | Viewed by 2375

Abstract

Solving phase equations for systems with high degrees of nonlinearities is cumbersome. However, in the case of two coupled canonical oscillators, that is, a reduced model of translated Wilson–Cowan neuronal dynamics, under slowly varying amplitude and rotating wave approximations, we suggested a convenient [...] Read more.

Solving phase equations for systems with high degrees of nonlinearities is cumbersome. However, in the case of two coupled canonical oscillators, that is, a reduced model of translated Wilson–Cowan neuronal dynamics, under slowly varying amplitude and rotating wave approximations, we suggested a convenient way to find their average relative phase evolution. This approach enabled us to find an explicit solution for the average relative phase of the two coupled canonical oscillators based on the original neuronal model parameters, and importantly, to find their phase-locking constraint. This methodology is straightforward to implement in any Wilson–Cowan-type coupled oscillators with applications in gradient frequency neural networks (GFNNs). Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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13 pages, 645 KiB

Open AccessArticle

Impact of a Multiple Pendulum with a Non-Linear Contact Force

by Dan B. Marghitu and Jing Zhao

Mathematics 2020, 8(8), 1202; https://doi.org/10.3390/math8081202 - 22 Jul 2020

Cited by 9 | Viewed by 3045

Abstract

This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain [...] Read more.

This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the non-linear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum the kinetic energy of the non-impacting link is increasing and the total kinetic energy of the impacting link is decreasing. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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18 pages, 7750 KiB

Open AccessArticle

An Efficient Analytical Approach to Investigate the Dynamics of a Misaligned Multirotor System

by Nicolae Herisanu and Vasile Marinca

Mathematics 2020, 8(7), 1083; https://doi.org/10.3390/math8071083 - 3 Jul 2020

Cited by 33 | Viewed by 1971

Abstract

This paper deals with the analytical investigation of a multirotor system connected by a flexible coupling subjected to dynamic angular misalignment. A novel analytical approach is employed for this purpose, namely the optimal auxiliary functions method, which proved to be successfully used to [...] Read more.

This paper deals with the analytical investigation of a multirotor system connected by a flexible coupling subjected to dynamic angular misalignment. A novel analytical approach is employed for this purpose, namely the optimal auxiliary functions method, which proved to be successfully used to obtain explicit analytical solutions to a system of strongly nonlinear differential equations with variable coefficients, useful in dynamic analysis of the considered multirotor system. The proposed procedure proved to be very efficient in practice for solving complicated nonlinear vibration problems. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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16 pages, 798 KiB

Open AccessArticle

Well-Posedness and Time Regularity for a System of Modified Korteweg-de Vries-Type Equations in Analytic Gevrey Spaces

by Aissa Boukarou, Kaddour Guerbati, Khaled Zennir, Sultan Alodhaibi and Salem Alkhalaf

Mathematics 2020, 8(5), 809; https://doi.org/10.3390/math8050809 - 16 May 2020

Cited by 10 | Viewed by 2304

Abstract

Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value [...] Read more.

Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value problem associated with a coupled system consisting of modified Korteweg-de Vries equations for given data. Furthermore, we prove that the unique solution belongs to Gevrey space

G^{σ} \times G^{σ}

in x and

G^{3 σ} \times G^{3 σ}

in t. This article is a continuation of recent studies reflected. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

13 pages, 4341 KiB

Open AccessArticle

Dynamics of the Stochastic Belousov-Zhabotinskii Chemical Reaction Model

by Ying Yang, Daqing Jiang, Donal O’Regan and Ahmed Alsaedi

Mathematics 2020, 8(5), 663; https://doi.org/10.3390/math8050663 - 27 Apr 2020

Cited by 2 | Viewed by 2448

Abstract

In this paper, we discuss the dynamic behavior of the stochastic Belousov-Zhabotinskii chemical reaction model. First, the existence and uniqueness of the stochastic model’s positive solution is proved. Then we show the stochastic Belousov-Zhabotinskii system has ergodicity and a stationary distribution. Finally, we [...] Read more.

In this paper, we discuss the dynamic behavior of the stochastic Belousov-Zhabotinskii chemical reaction model. First, the existence and uniqueness of the stochastic model’s positive solution is proved. Then we show the stochastic Belousov-Zhabotinskii system has ergodicity and a stationary distribution. Finally, we present some simulations to illustrate our theoretical results. We note that the unique equilibrium of the original ordinary differential equation model is globally asymptotically stable under appropriate conditions of the parameter value f, while the stochastic model is ergodic regardless of the value of f. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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8 pages, 749 KiB

Open AccessArticle

A New Approach in the Study of Oscillation Criteria of Even-Order Neutral Differential Equations

by Osama Moaaz, Jan Awrejcewicz and Omar Bazighifan

Mathematics 2020, 8(2), 197; https://doi.org/10.3390/math8020197 - 5 Feb 2020

Cited by 40 | Viewed by 2308

Abstract

Based on the comparison with first-order delay equations, we establish a new oscillation criterion for a class of even-order neutral differential equations. Our new criterion improves a number of existing ones. An illustrative example is provided. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

26 pages, 4031 KiB

Open AccessArticle

A New Nine-Dimensional Chaotic Lorenz System with Quaternion Variables: Complicated Dynamics, Electronic Circuit Design, Anti-Anticipating Synchronization, and Chaotic Masking Communication Application

by Emad E. Mahmoud, M. Higazy and Turkiah M. Al-Harthi

Mathematics 2019, 7(10), 877; https://doi.org/10.3390/math7100877 - 20 Sep 2019

Cited by 25 | Viewed by 4120

Abstract

In this paper, a chaotic quaternion autonomous nonlinear structure is introduced and intends to be a contribution. It is the first nonlinear dynamical system with quaternion variables to be studied in the literature. With nine dimensions, the new system is a high-dimensional one. [...] Read more.

In this paper, a chaotic quaternion autonomous nonlinear structure is introduced and intends to be a contribution. It is the first nonlinear dynamical system with quaternion variables to be studied in the literature. With nine dimensions, the new system is a high-dimensional one. Several vital characteristics and features of this model are investigated, such as its Hamiltonian, symmetry, signal flow graph, dissipation, equilibriums and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams, and chaotic behavior. A circuit implementation is designed to realize the new system, and a scheme is designed to achieve anti-anticipating synchronization (AAS) of two identical chaotic attractors with quaternion variables based on a Lyapunov function and active control. The concept of AAS is yet to be explored in the literature. A simulation experiment is designed and executed to illustrate the effectiveness of the acquired results. After synchronization, numerical outcomes are planned to explain the status variables and errors of these chaotic attractors to prove that AAS is achieved. The secure communication problem is studied based on the obtained events of the AAS of two identical nonlinear Lorenz systems with quaternion variables. AAS connecting the drive and response systems in chaotic systems with quaternion variables is the key to achieving communication. Signal encryption and restoration are simulated numerically. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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