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The present research focuses on the challenges engineers face in predicting the behavior of nonlinear vibration systems accurately. The nonperturbative method is highlighted as a solution that provides insights into chaos, bifurcation, resonance response, and stability attributes. Specifically, the study delves into the dynamic analysis of the nonlinear Mathieu equation. The research involves a complex and extensive analytical exploration, transitioning from a nonlinear state to a linear one through various stages. The introduced computational method aims to examine the resonance response of the nonlinear Mathieu equation and offer innovative solutions for the Mathieu–Duffing–type oscillator. The nonperturbative approach remains essential in gaining a deeper understanding of nonlinear vibration systems. [ABSTRACT FROM AUTHOR]

The frequency-amplitude relationship is pivotal in understanding oscillatory systems, dictating their dynamic behaviors and responses. This paper delves into the intricacies of the frequency amplitude formula, elucidating its foundational role in both linear and nonlinear oscillations. Through comprehensive analysis, we highlight the formula's significance in predicting system behaviors, especially in environments with varying amplitudes. Our findings underscore the formula's potential as a robust tool for enhanced system characterization, offering profound implications for diverse applications across scientific and engineering domains. This study delves deep into the formulation of the frequency-amplitude relationship, extending its application beyond the conventional cubic powers of the restoring force. We introduce three novel and equivalent formulations of the generalized frequency amplitude, breaking traditional boundaries by not confining the restoring force to just odd functions. The formula to determine the frequency of the singular oscillator has been set forth. Our approach, characterized by its simplicity, offers a robust tool for analyzing high nonlinearity vibrations, ushering in a richer, multidimensional perspective to vibration analysis. [ABSTRACT FROM AUTHOR]

The current research studies a fractal Duffing oscillator in the presence of periodic force. To find an analytic solution for this oscillator, the aspects explained in the following are considered. First, we obtain an alternative unforced fractal fourthorder equation and then convert it into a continuous space. Therefore, the nonperturbative (NP) approach is used to calculate the analytic solution for the alternate equation in the second-order form after reducing its rank. It is seen that the analytical and numerical solutions agree very well. The computations reveal that for every value of the fraction parameter, the approximation and numerical solutions are identical. The present study gives reliability in the technique of reducing the order of differential equations. Furthermore, the required periodic solution is also obtained by Galerkin’s technique. In contrast to the traditional technique, which works to transform the variable and is valid only in the absence of external forces, if there is an external force, it leads to significant mathematical difficulties. The current technique works on the operator, which is simple and effective when investigating fractal oscillators with external forces, easy to obtain analytic solutions, and doesn't lead to any mathematical difficulties. [ABSTRACT FROM AUTHOR]

In this brief communication, we highlight several critical inaccuracies in the paper entitled "A novel methodology for a time-delayed controller to prevent nonlinear system oscillations," identified by the DOI: 10.1177/14613484231195276. The core of the non-perturbative approach discussed in this paper is the introduction of a trial solution with an indeterminate nonlinear frequency. We elucidate the technical missteps and provide necessary corrections. The scrutinized article presents ample opportunities for enhancement, and we propose several amendments. Additionally, we reveal that the numerical outcomes showcased in the mentioned paper expose deficiencies in the proposed approximate solution. With the amendments implemented herein, a more precise numerical solution is achieved. [ABSTRACT FROM AUTHOR]

The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu–Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu–Duffing equation in the continuous space by employing a new modification of He's definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, "the non-perturbative approach", that deals with the obtained equation has been introduced. [ABSTRACT FROM AUTHOR]

In the present study, several successive approximate solutions of the nonlinear oscillator are derived by using the efficient frequency formula. A systematical analysis of the formulation of the nonlinear frequency helps to establish a general periodic solution. Each approximation represents, individually, the solution of the nonlinear oscillator. For the optimal design and accurate prediction of structural behavior, a new optimizer is demonstrated for efficient solutions. The classical Duffing frequency formula has been modified. The numerical calculations show high agreement with the exact frequency. The justifiability of the obtained solutions is confirmed by comparison with the numerical solution. It is shown that the enhanced solution is accurate for large amplitudes and is not restricted to oscillations that have small amplitudes. The new approach can provide a perfect approximation for the nonlinear oscillation. [ABSTRACT FROM AUTHOR]

The impetus for the current investigation relates to the implementation of an efficient novel technique to obtain a time-delayed vibration control analytical solution. The current technique follows simple and easy-to-apply criteria. This technique is based on converting the nonlinear time-delayed Van der Pol (VDP)-Duffing oscillator to an equivalent linear one. Details of the conversion to an equivalent linear ordinary differential equation are mentioned. The convergence between the numerical outcomes and the analytic solution is achieved and gives a satisfying accuracy of the equivalent result. [ABSTRACT FROM AUTHOR]

An immediate analytic solution for the time-delayed fractional linear jerk oscillator is derived in the present study. The currently used method adheres to basic, easily implementable criteria. This proposed approach is based on transforming the time-delayed fractional oscillator to an equivalent oscillator with ordinary derivatives, which makes the solution procedure simple. The conversion process steps to an equivalent ordinary differential oscillator are displayed. Galerkin's method is used to derive the jerk frequency. The good match between the analytic and numerical outcomes shows that the equivalent linearized mechanism leads to finding a satisfying and accurate solution. Stability discussion is conducted analytically and numerically. [ABSTRACT FROM AUTHOR]

Numerous theoretical physics and chemistry problems can be modeled using Mathieu's equations (MEs). They are crucial to the theory of potential energy in quantum systems, which is equivalent to the Schrödinger equation. According to the mentioned applications, thus, the current study investigates the stability behavior of the nonlinear-coupled MEs. The analysis of the coupled harmonic resonance cases imposes two coupled solvability conditions, which leads to coupled parametric nonlinear Landau equations. In addition, a super-harmonic nonlinear resonance combination is presented. Solutions and stability criteria are discussed for each case. It is shown that resonance produces an unstable system. The transition curves are derived. Numerical calculations show the excitation of the frequency on the periodic solutions. [ABSTRACT FROM AUTHOR]

The current paper presents the first treatment to obtain the solution of a complex Helmholtz–Duffing oscillator. This model has described the elevation of the surface waves in fluid mechanics. The non-perturbative approach is used to convert the nonlinear oscillator to a linear oscillator to derive the solution of the complex nonlinear oscillator. This paper has opened the path for a new way to study more complex nonlinear elevation of surface waves. [ABSTRACT FROM AUTHOR]

The current article is concerned with a comprehensive investigation of achieving the simplest solution of non-conservative coupled nonlinear forced oscillators. The article mainly depends on a non-perturbative method. It depends on yielding an equivalent linear system. The advantage of this linear system is that its coefficients are easily computed and that they include the effects of the original nonlinear coefficients. This linearization approach allows achieving quasi-exact solutions even in the presence of periodic forces. The relationships of frequencies with amplitudes are easily established. The analytical solution so yielded may serve as a basis for a qualitative understanding of the actual behavior of the coupled nonlinear oscillators. Additionally, numerical calculations are carried out graphically to address the validation of the new approach and to further illustrate the effectiveness and convenience of the method. The results are compared with the exact numerical solutions which show perfect accuracy. The method can be easily extended to other nonlinear systems and can therefore be exceedingly applicable in engineering and other sciences. [ABSTRACT FROM AUTHOR]

The comprehensive formularization of the frequency–amplitude formula is the main interest in this work to cover higher powers of the restoring force which are not limited to cubic powers. Three equivalent styles of the generalized frequency amplitude have been performed. In addition, the restoring force is not restricted to an odd function in which the non-secular forces are included. Three forms of the non-secular forces have been formulated for the first time and treated as non-homogenous of the linearized Duffing oscillator. The modified style of He's formula has been applied to the singular oscillator using the enhanced potential function. The simplicity of the present approach provides extra advantages for high nonlinearity vibration. This method enriches the analysis with more details in new dimensions. [ABSTRACT FROM AUTHOR]

The existence of the derivative with a fractional-order in a class of differential equations could lead to complicating the analysis. In this paper, a novel approach has been introduced to facilitate the analysis of the oscillation having fractional-order derivatives and to obtain the analytical solution easily. The present technique is formulated to provide an easy way of understanding. The suggested technique has been utilized to study two examples for illustration. The similitude between the analytical and numerical solution verifies and gives satisfactory precision to the equivalent solution. The new technique is represented to be the best tool for solving the nonlinear oscillation problems in physics and engineering which have a fractional-order. [ABSTRACT FROM AUTHOR]

The vibration of a regular rigid bar without sliding over a solid annular surface of a specified radius can be considered by a parametric Gaylord's oscillator. The governing equation was the result of a strong nonlinear oscillation without having a natural frequency. The present work is concerned with obtaining the approximate solution and amplitude-frequency equation of the parametric Gaylord's equation via an easier process. The non-perturbative approach was applied twice to analyze the present oscillator. Two steps are used, the first is to transform Gaylord's oscillator to the parametric pendulum equation having a natural frequency. The second step is to establish the amplitude-frequency relationship which was taken out in terms of the Bessel functions. A periodic analytic solution is obtained, in the presence or without the parametric force. The frequency at the resonance case is established without a perturbation for the first time. The stability condition is established and discussed graphically. The analytic solution was also validated by comparing it with its corresponding numerical data which showed a very good agreement. In a word, by dissection of the behavior of strong nonlinearity oscillators, the non-perturbative technique is characterized by its ease and simplicity along with high accuracy when compared to other perturbative methods. [ABSTRACT FROM AUTHOR]

Nonlinear differential equations that control the propagation of a surface wave through the surface disconnection between two fluids are described by the Helmholtz–Duffing oscillator having imaginary damping forces. This oscillator is solved without using any perturbation techniques. This study is relevant in many fields such as nanotechnology. Along with the nonlinear analysis, the periodic solution and the stability criteria are established. Numerical calculations for stability conditions showed vital changes in the stability behavior due to the presence of the rotation ratio. [ABSTRACT FROM AUTHOR]

The Gaylord's oscillator is a vibrating of a uniform rigid rod without slipping on a rigid circular surface with a definite radius. The dominant equation of motion was the outcome of a strongly nonlinear pendulum equation of the second order. The run article is interested in obtaining the frequency–amplitude equation and the approximate solution of Gaylord's oscillator by a simpler approach. The frequency–amplitude relationship is derived in terms of the Bessel function. Quasi-exact periodic solution derived depends on a non-perturbative approach. The validation of the analytical solution is made through the comparison with the numerical solution which shows excellent approval. Finally, the non-perturbative method is of high accuracy besides simplicity if it is compared with the other perturbative techniques in analyzing the behavior of oscillators with strong nonlinearities. [ABSTRACT FROM AUTHOR]

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He's frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He's ideas and the present authors' previous work on the HPM, so the citation might not be exhaustive. [ABSTRACT FROM AUTHOR]

This paper is devoted to obtain an approximate solution to the damped quintic–cubic nonlinear Duffing–Mathieu equation via a modified homotopy perturbation method (HPM). The modification under consideration deals with the improvement of the HPM with the exponential decay parameter. This scheme allows us to get a solution to the damped nonlinear Duffing–Mathieu equation, which the classical HPM failed to obtain. It is found that the solutions and the characteristic curves are affected by the presence of the damping force. The frequency-amplitude characteristics of a symbiotic solution are confirmed as well as the stability condition is carried out in the (non)-resonance cases. All the calculations are done via Mathematica. The comparison between both of the numerical and analytical solutions showed a very good agreement. Illustrated graphs are plotted for a superior realization of periodic motions in the Duffing–Mathieu oscillator. Nonlinear behaviors of each oscillation motion have been characterized through frequency curves. [ABSTRACT FROM AUTHOR]

The vibration theory has become a hot topic in both mathematics and engineering. This work is an attempt to reduce the complexity of the Toda model and to exploit the advantages of annihilator operators. The reducing rank method is proposed for the outcoming jerk oscillator model. The modified homotopy perturbation method (HPM) is presented to solve the resulting model. Besides, to simplify the model, the outstanding features of the resonance reaction are preserved. It is declared that the obtained results are robust as they can exist in a wide range of oscillators. It is found that the amplitude of oscillation greatly affects its periodic nature and has an exponential form. The dynamics of a non‐autonomous oscillator in which the frequency of the external force depends on the dynamical variable is studied. It is also revealed that the linear damping coefficient plays a dual role in the stability behavior, while the nonlinear counterpart exerts a destabilizing influence on the resonance response. [ABSTRACT FROM AUTHOR]

The current paper is concerned with the nonlinear stability analysis of rotating magnetic fluid columns. The rotation sources are a mixture of both uniform and oscillating behavior. The motivation behind tackling this topic is the increasing interest in atmospheric and oceanic motions. The system consists of two magnetic phase fluid that fills two infinite vertical cylinders. An azimuthal uniform magnetic field is penetrated on the system. The governing equations of motion, in terms of the Coriolis force and reduced pressure, along with Maxwell's equation in the quasi-static approximations are considered. Consequently, the disturbance of the interface has an azimuthal behavior. The fluids are fully saturated in porous media. In light of the implication of the nonlinear boundary conditions, the solutions of the linearized equations of motion resulted in a nonlinear characteristic dispersion equation. Utilizing the homotopy perturbation technique, this equation is analyzed. A modification of the latter equation is made to seem like a nonlinear Klein–Gordon equation. The stability criteria are realized in linear as well as nonlinear approaches. A set of diagrams is graphed to illustrate the effects of several non-dimensional numbers on the stability profile in resonance as well as non-resonance cases. [ABSTRACT FROM AUTHOR]

Control forces of delayed third-order critically damped Duffing equation is proposed in this study. The analytical solution is based on the modified HPM. The solution is demonstrated by introducing a proper function used to suppress the vibration of the nonlinear oscillations. The outcome of the modified homotopy expansion by the exponential negative delay parameter reveals that approximations using this technique are more accurate. To test the accuracy of the recommended modified technique, the obtained results are compared with the numerical solution. Excellent agreement with numerical testing is observed. Further, the suggested control function is eligible for establishing highly accurate approximate analytical frequency and periodic solutions. Based on He's frequency formula a quasi-exact solution for the nonlinear oscillation has been derived. It is effective and simple to utilize analytic tools for nonlinear problems. These results reveal the accuracy and quality of the suggested technique. [ABSTRACT FROM AUTHOR]

In this article, the main objective is to employ the homotopy perturbation method (HPM) as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients. As a simple example, the nonlinear damping Mathieu equation has been investigated. In this investigation, two nonlinear solvability conditions are imposed. One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the firstorder solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science. [ABSTRACT FROM AUTHOR]

In the present work, an enhanced perturbation analysis to solve a time-fractional Klein–Gordon equation (KG equation) and obtain an analytic approximate periodic solution is examined. The Riemann–Liouville fractional derivative is utilised. A travelling wave solution is adopted throughout the perturbation method by including two small perturbation parameters. The amplitude equation is formulated in the form of a cubic–quintic complex nonlinear Schrödinger equation. The solution of this equation leads to a transcendental frequency equation. An approximate solution to this frequency equation is performed. The stability criteria are derived. The procedure adopted here is very significant and powerful for solving many nonlinear partial differential equations (NLPDEs) arising in nonlinear science and engineering. [ABSTRACT FROM AUTHOR]

In this paper, a tutorial introduction to the two-scale fractal calculus is given. The two-scale fractal derivative is conformable with the traditional differential derivatives. When the fractal dimensions tend to an integer value, its basic properties are discussed, and the fractal Zhiber–Shabat oscillator is used as an example to reveal the basic properties of a fractal differential equation. The two-scale transform is used to convert the nonlinear Zhiber–Shabat oscillator with the fractal derivatives to the traditional model. The homotopy perturbation method has been demonstrated under a suitable transformation of the system containing several exponential nonlinear terms to the famous Helmholtz–Duffing oscillator. Stability behavior is discussed. Several numerical illustrations are also provided to exhibit the integrity of the introduced formulation. It is demonstrated that the proposed formulation is accurate enough for highly nonlinear differential equations containing large nonlinear terms. [ABSTRACT FROM AUTHOR]

A governing equation is established for string axial vibrations with temporal and spatial damping forces by the Hamilton principle. It is an extension of the well-known Klein-Gordon equation. The classical homotopy perturbation method (HPM) fails to analyze this equation, and a modification with an exponential decay parameter is suggested. The analysis shows that the amplitude behaves as an exponential decay by the damping parameter. Furthermore, the frequency equation is established and the stability condition is performed. The modified homotopy perturbation method yields a more effective result for the nonlinear oscillators and helps to overcome the shortcoming of the classical approach. The comparison between the analytical solution and the numerical solution shows an excellent agreement. [ABSTRACT FROM AUTHOR]

This paper shows that the Toda oscillator can be converted to a new Helmholtz-Fangzhu equation, which is solved by the homotopy perturbation method with three expansions. The solution, the negative frequency and the damping rate are expanded into power forms in the homotopy parameter, the solution process is given step by step, and the results are of high accuracy. This paper gives a general approach to damp vibrations with singular terms,especially to the Helmholtz-Fangzhu oscillator. [ABSTRACT FROM AUTHOR]

The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically. [ABSTRACT FROM AUTHOR]

Many nonlinear vibrations arising in the engineering of architecture include noise and uncertain properties, and this paper suggests a fractional model to elucidate the properties. The fractional sine-Gordon equation with the Riemann–Liouville fractional derivative is used as an example to solve its periodic solution by the homotopy perturbation method. The frequency–amplitude relationship is obtained, and the effect of the fractional derivative order on the vibration property is discussed. Additionally, the harmonic resonance is also discussed. This preliminary research can be further extended to real applications. [ABSTRACT FROM AUTHOR]

A modification of the homotopy perturbation method is suggested with three effective expansions to solve a nonlinear oscillator with damping terms to expand the solution, the frequency and the amplitude. The Duffing equation with linear damping is used as an example to illustrate the simple solution process and effective results. The analysis exhibits that the amplitude behaves as an exponential decay with the damping parameter. This scheme yields a more effective result for the nonlinear oscillators and overcomes the shortcoming in some problems. [ABSTRACT FROM AUTHOR]

In the current work, we apply a nonstandard scheme to solve the third‐order Duffing equation. This equation is produced from the strong damped Klein–Gordon equation under the traveling wave transformation. The solution and the stability conditions for the third‐order Duffing equation have been discussed for the first time. The present analysis is new and used the reducing rank method with the homotopy perturbation method. A nonoscillator solution with the oscillating solutions is derived individually and frequency formula is obtained. Besides, the stability analysis is discussed. [ABSTRACT FROM AUTHOR]

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution. [ABSTRACT FROM AUTHOR]

An accurate frequency-amplitude relationship is very needed to elucidate the properties of the oldest device of Fangzhu for collecting water from the air. The Fangzhu oscillator was derived and solved approximately (He et al. in Math Methods Appl Sci, 2020, 10.1002/mma.6384), here we show that the singular Duffing-like oscillator can be more effectively solved by the homotopy perturbation method and a criterion is obtained for the existence of a periodic solution for the singular differential equation. The results obtained in this paper are helpful for the optimal design of the Fangzhu device. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear instability of an interface between two magnetic fluids separated by a cylindrical interface in porous media. The system is influenced by a uniform axial magnetic field. The magnetic field intensities allow a presence of surface currents at the interface. The transfer of mass and heat across the interface is considered. The solutions of linearized equations of motion, under the appropriate nonlinear boundary conditions, lead to a nonlinear characteristic equation that is governed the behavior of the interface deflection. Drawing on the linear stability theory, Routh–Hurwitz's criteria are utilized to judge the stability criteria. The coupling of Laplace transforms and Homotopy perturbation techniques are adopted to obtain an approximate analytical solution of the interface profile. The nonlinear stability analysis resulted in two levels of solvability conditions. By means of these conditions, a Ginzburg–Landau equation is conducted. The latter equation represented the nonlinear stability configuration. The magnetic field intensity was plotted versus the wave number of the surface waves. Therefore, the stability picture was divided into stable and as well as unstable regions. Subsequently, the influence of the various physical parameters was addressed. [ABSTRACT FROM AUTHOR]

In the present proposal, the familiar method of the parameter expansion is combined with the multiple scales to study the stability behaviour of the Riemann–Liouville fractional derivative applied to the cubic delayed Duffing oscillator. The analysis of the modified multiple scale perturbation leads to a system of nonlinear differential-algebraic equations governing the solvability conditions. The nonlinear differential equation was reduced to the linear differential equation with the help of the algebraic one. The stability attitude of the periodic motion is determined by the steady-state analysis. Such a periodic motion is needed to better understand the dynamics of the fractional cubic delayed Duffing oscillator. [ABSTRACT FROM AUTHOR]

The current paper is concerned with the motion of a rocking uniform rigid rod, without slipping, over a rigid circular surface. The governing equation of motion resulted in a highly nonlinear second-order ordinary differential equation. This nonlinear equation has no natural frequency. A coupling the homotopy perturbation method and Laplace transform is adopted to obtain an approximate solution of the equation of motion. In addition, He's transformation method is used to obtain a periodic solution. Stability conditions are derived by making use of a nonlinear frequency analysis. Numerical calculations are achieved to investigate the governed perturbed solutions as well as the stability picture. It is found that the radius of the length of the rigid rod as well as the radius of the circular surface has a stabilizing influence. In contrast, the initial angular velocity has a destabilizing effect. [ABSTRACT FROM AUTHOR]

In this paper, we perform the frequency-expansion formula for the nonlinear cubic damping van der Pol's equation, and the nonlinear frequency is derived. Stability conditions are performed, for the first time ever, by the nonlinear frequency technology and for the nonlinear oscillator. In terms of the van der Pol's coefficients the stability conditions have been performed. Further, the stability conditions are performed in the case of the complex damping coefficients. Moreover, the study has been extended to include the influence of a forcing van der Pol' oscillator. Stability conditions have been derived at each resonance case. Redoing the perturbation theory for the van der Pol oscillator illustrates more of a resonance formulation such as sub-harmonic resonance and super-harmonic resonance. More approximate nonlinear dispersion relations of quartic and quintic forms in the squaring of the extended frequency are derived, respectively. [ABSTRACT FROM AUTHOR]

Two dimensions Mathieu equation containing periodic terms as well as the delayed parameters has been investigated in the present work. The present system represents to a generalized form of the one-dimension delay Mathieu equation. The mathematical difficulty for delay the coupled Mathieu equation has been overcome by using the matrices method. Properties of inverse complex matrices enable us to transform the vector form of the solvability conditions to the scalar form. Small oscillation about a marginal state is introduced by using the method of multiple scales. Stability criteria for the complex matrices have been established and lead to obtain resonance curves. The analysis has been extended so that the delay 2-dimensions Mathieu equation containing weak complex damping part. Stability conditions and the transition curves that included the in uence of both the delayed as well the complex damping terms has been obtained. The transition curves are analyzed using the method of harmonic balance. We note that the delayed higher dimension of the parametric excitation has a great interest and application to the design of nuclear accelerators. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for studying the nonlinear Rayleigh–Taylor instability (RTI). The system deals with two rotating superposed infinite hydromagnetic Darcian flows through porous media under the influence of a uniform tangential magnetic field. The field allows the presence of currents on the surface of separation. The appropriate linear governing equations are solved and confirmed with the corresponding nonlinear boundary conditions. A nonlinear characteristic of the surface deflection is deducted. Away from the traditional techniques of the stability analysis, the work introduces a new one. The analysis depends mainly on the homotopy perturbation method (HPM). To achieve an analytical approximate periodic solution of the surface deflection, the secular terms are removed. This cancellation resulted in well-known amplitude equations. These equations are utilised to achieve stability criteria of the system. Therefore, the stability configuration is exercised in linear as well as nonlinear approaches. The mathematical procedure adopted here is simple, promising and powerful. The method may be used to treat more complicated nonlinear differential equations that arise in science, physics and engineering applications. A numerical calculation is performed to graph the implication of various parameters on the stability picture. In addition, for more convenience, the surface deflection is depicted. [ABSTRACT FROM AUTHOR]