Secular Terms Research Articles

On the Sitnikov‐like N ‐body problem with quasi‐homogeneous potential

In the present article, the periodic solutions of the N-body with quasi-homogeneous potential in the Sitnikov sense by applying the multiple methods of scale (MMS) and Lindstedt–Poincaré (LP) technique are obtained. However, these methods are used to find the approximate periodic solutions in the closed form by eliminating the secular terms. In addition of the Newtonian potential and forces, we consider that the big bodies create quasi-homogeneous potentials. We add the inverse cubic corrective term to the inverse square Newtonian law of gravitation, in order to approximate the various phenomena due to the shape of the bodies or the radiation emitting from them. We study the Sitnikov motion in the N-bodies under this consideration. We, further, analyzed the obtain approximate periodic solutions of the Sitnikov motion, for ν = 2 , 7 by using the MMS and LP-method, in closed form. The numerical comparisons are presented in the first and second approximated solutions obtained by using MMS and numerical solutions obtained by LP-method are illustrated graphically. The effect of initial conditions on the solutions of the Sitnikov motion is illustrated graphically obtained by both the techniques. It is observed that the choice of initial conditions plays a crucial role in the numerical and approximate solutions.

The study of the nonlinear vibration of S-S and C-C laminated composite conical shells on elastic foundations through an approximate method

This paper investigates the nonlinear vibration responses of laminated composite conical shells surrounded by elastic foundations under S-S and C-C boundary conditions via an approximate approach. The laminated composite conical shells are modeled based on classical shell theory of Love employing von Karman nonlinear theory. Nonlinear vibration equation of the conical shells is extracted by handling Lagrange method. The linear and nonlinear vibration responses are obtained via an approximate method which combines Lindstedt-Poincare method with modal analysis. The validation of this study is carried out through the comparison of the results of this study with results of published literature. The effects of several parameters including the constants of elastic foundations, boundary conditions, total thickness, length, large edge radius and semi-vertex angle on the values of fundamental linear frequency and curves of amplitude parameter versus nonlinear frequency ratio for laminated composite conical shells with both S-S and C-C boundary conditions are investigated.

Reduced Nonlinear Model for Orbit Uncertainty Propagation and Estimation

This paper presents a novel method for nonlinear uncertainty propagation and estimation in orbital dynamics. The proposed technique relies on a Taylor series expansion of the integral flow to model the dynamics around the reference solution and introduces an approximation of the high-order variational equations that reduces the complexity of evaluating the series. In particular, the high-order state-transition tensors (STTs) are approximated by capturing the dominant secular terms. Simple expressions to compute them are provided. The approximation stems from confining the Lyapunov instability of the motion to the time domain. The result is a time-explicit approximation of the STTs that can be used to predict the evolution of the uncertainty distribution accounting for nonlinear effects with minimal overhead. Finally, a high-order version of the extended Kalman filter is developed by implementing the approximation of the nonlinear terms of the Taylor series into an estimation scheme. The performance of the algorithm is evaluated with several practical examples.

Approximate Solutions of Coupled Nonlinear Oscillations: Stability Analysis

The current article is concerned with a comprehensive investigation in achieving approximate solutions of coupled nonlinear oscillations with high nonlinearity. These equations are highly nonlinear second-order ordinary differential equations. Via a coupling of the Homotopy perturbation method and Laplace transforms, which is so-called the He-Laplace method, traditional approximate solutions involving the secular terms are accomplished. On the other hand, in order to cancel the secular terms, an expanded frequency technique is adapted to accomplish periodic approximate solutions. Therefore, a nonlinear frequency, for each differential equation, is achieved. Furthermore, for more convenience, these solutions are pictured to indicate their behavior. The multiple time-scales with the aid of the Homotopy concept are utilized to judge the stability criteria. The analyses reveal the resonance as well as the non-resonant cases. Additionally, numerical calculations are carried out, graphically, to address the regions that guaranteed the bounded solutions. It is found that the latter method, is the most powerful mathematical tool in extracting the stability analysis of the considered system.

Station-keeping error analysis for halo orbits around Libration point L1 using linear control logic

In this article, we perform the computation of halo orbits around libration point L1 in the Sun–Jupiter system. Third-order approximation, Lindstedt–Poincare method is used to select the initial guess for libration point orbits. Furthermore, Newton’s differential correction scheme is utilized to correct the initial guess obtained from an analytical approach. Halo orbits are exceptionally unstable, the station-keeping maneuver is required to maintain the nominal halo orbit in the vicinity of collinear libration points. We have applied linear control logic in the non-linear equation of motion for the station-keeping error analysis of halo orbits and also computed the control input for halo orbits. Numerical results illustrate the viability of the proposed control technique for the exact station-keeping error at libration point L1.

Decolonial Translation: Destabilizing Coloniality in Secular Translations of Islamic Law

Abstract Contemporary Islamic legal studies – both inside and outside the Muslim world – commonly relies upon a secular distortion of law. In this article, I use translation as a metonym for secular transformations and, accordingly, I will demonstrate how secular ideology translates the Islamic tradition. A secular translation converts the Islamic tradition into “religion” (the non-secular) and Islamic law into “sharia” – a term intended to represent the English mispronunciation of the Arabic word شريعة (sharīʿah). I explore the differences between historical Islamic terms and secular terms in order to demonstrate that coloniality generates religion and religious law; in turn, these two notions convert شريعة (sharīʿah) into “sharia” in both Arabic and non-Arabic languages. Consequently, the notion of “sharia” is part of a colonial system of meaning.

Sitnikov five-body problem with combined effects of radiation pressure and oblateness

We study the motion of negligible mass in the frame work of Sitnikov five-body problem where four equal oblate spheroids known as primaries symmetrical in all respect are placed at the vertices of square. These primaries are also considered as source of radiations moving in a circular orbit around their common center of mass. The fifth negligible mass performs oscillations along z-axis which is perpendicular to the orbital plane of motion of the primaries and passes through the center of mass of the primaries. Under the combined effects of radiation pressure and oblateness, we have developed the series solution by the Lindstedt-Poincare technique and established averaged Hamiltonians by applying the Van der Pol transformation and averaging technique of Guckenheimer and Holmes (1983). The orbits such as regular, periodic, quasi-periodic, chaotic, or stochastic have been examined with the help of Poincare surfaces of section.

El discurso herético en Prosas Profanas: ¿secularización o profanación?

Prosas Profanas y otros poemas by Rubén Darío is built from its initial paratext in a polyphonic way. The title of this collection of poems is based on a negation: the word “profane” is etymologically defined as the negation of religion, which can be understood both in secular and heretical terms. Since all negations imply the incorporation of another’s discourse to be denied, it is possible to affirm that this title encloses the incorporation of a counter-discourse of Christianity. “Ite, missa est” and “El reino interior” are two poems that show how specific genres of the Catholic religion (the mass and the narrative iconography) are subverted to create their own denial. The dialogical relationship between religious discourse and their heretical counter-discourses occurs in Prosas Profanas upon the creation of effects of généricité, since Rubén Darío plays with the limits of highly codified discursive genres. The aim of the following work is to describe how discursive heterogeneity operates as a matrix behind the construction of Prosas Profanas, that is, how it relies on religious discourse in terms of opposition in order to build new meanings of a heretical nature.

Water Oscillation Modelling Inside OWC Chamber for Wave Energy Conversion

Among wave energy converters (WECs), the Oscillating Water Column (OWC) system is considered as one of the most promising converters; this system works on the principle of water oscillations in an enclosed air chamber due to incident sea waves. The amount of pneumatic power that can be harvested depends on the elevation level, the water oscillations' frequency inside the chamber, and the chamber's global damping. A one-dimensional (1-D) unsteady model has been elaborated to analyze the water elevation's dynamic behavior. Great attention has been paid to determining the natural frequency and the global damping for maximum wave energy capture. The model is based on a water mass block moving periodically upward and downward as a piston inside the chamber, inducing the enclosed air to flow alternately through a turbine. The Lagrangian formalism has been adopted to derive the governing differential equation for the piston water motion. Then the poincare-Lindstedt method has been used to deal with the non-linearity of the problem. The elaborated model has been adapted to an OWC system in the case of monochromatic waves. Results have been compared to another referenced paper for validation and pertinence. Fair agreements have been noted.

Rotational Dynamics of the Inner Satellites of Jupiter

The article considers the attitude stability of synchronous rotation and the most significant relativistic effects in the rotational dynamics of the inner satellites of Jupiter: Metis (J16), Adrastea (J15), Amalthea (J5), and Thebe (J14). It is established that the plane synchronous rotation of all inner satellites of Jupiter for the most probable values of the parameters of their shapes is stable with respect to tilting the axis of rotation. For the first time, the most significant secular, periodic, and mixed terms of the geodetic rotation of the inner satellites of Jupiter in the Euler angles relative to their own coordinate systems and in the angles of their rotation with respect to the fixed equator of the Earth and the vernal equinox (for the J2000.0 epoch) are determined. It is shown that there are objects in the Solar System with significant geodetic rotation caused primarily by their proximity to the perturbing central body rather than its mass. In particular, the value of the geodetic precession of the inner satellites of Jupiter (for which Jupiter is a less massive perturbing central body than the Sun) is 105 times greater than that of Jupiter rotating around its more massive central body (the Sun) and comparable with their precession in Newton approximation.

Applications of the Differential Transformation Method and Multi-Step Differential Transformation Method to Solve a Rotavirus Epidemic Model

Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S; V; I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods

Analysis and prediction of shock formation in acoustic energy transfer systems

Losses associated with nonlinear wave propagation and exhibited by acoustic wave distortion and shock formation compromise the efficiency of contactless acoustic energy transfer systems. As such, predicting the shock formation distance and its dependence on the amplitude of the excitation is essential for their efficiency, design, and operation. We present an analytical approach capable of predicting the shock formation distance of acoustic waves generated by a baffled disk with arbitrary deformation in a weakly viscous fluid medium. The lossless Westervelt equation, used to model the nonlinear wave propagation, is asymptotically expanded based on the amplitude of the excitation. Because the solutions of the first- and second-order equations decay at different rates, we implement the method of renormalization and introduce a coordinate transformation to identify and eliminate the secular terms. The approach yields two partial differential equations that can be solved to predict the formation distance either analytically or numerically much faster than time-domain numerical simulations. The analysis and results are validated with solutions obtained from a nonlinear finite element simulation and previous experimental measurements.

The modified Lindstedt–Poincare method for solving quadratic nonlinear oscillators

Recently, an analytical solution of a quadratic nonlinear oscillator has been presented based on the harmonic balance method. By introducing a small parameter, a set of nonlinear algebraic equations have been solved which usually appear among unknown coefficients of several harmonic terms. But the method is not suitable for all quadratic oscillators. Earlier, introducing a small parameter to the frequency series, Cheung et al. modified the Lindstedt–Poincare method and used it to solve strong nonlinear oscillators including a quadratic oscillator. But due to some limitations of both parameters, a changed form of frequency-related parameter (introduced by Cheung et al.) has been presented for solving various quadratic oscillators.

A perturbation analysis in nonlinear free vibration of a microbeam reinforced by shape memory alloy layer based on the modified couple stress theory

Numerous applications have been found in various microelectromechanical systems for shape memory alloys. This study analyzes the nonlinear free vibrations of a sandwiched microbeam, consisting of a shape memory alloy core and two elastic layers on either side. The behavior of the shape memory layer is modeled with the one-dimensional Brinson model. The equation of motion is derived from Hamilton’s principle and the modified couple stress theory for the Euler–Bernoulli beam and then truncated into a reduced-order model through Galerkin’s technique. An approximate analytical solution is obtained using the Lindstedt–Poincare method for different temperatures. The effects of temperature, material length scale parameter, aspect ratio, and maximum vibration amplitude on nonlinear normalized frequency have been investigated. The results have been compared with those of some previous studies and can provide a better understanding of the design of sandwiched microbeams, which have a shape memory alloy layer.

A remark on renormalization group theoretical perturbation in a class of ordinary differential equations

Abstract We revisit the renormalization group (RG) theoretical perturbation theory on oscillator-type second-order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all orders of the RG series. The result covers nonautonomous as well as autonomous cases and refines earlier studies, including the classic examples of Van der Pol, Mathieu, Duffing, and Rayleigh equations.

Second-order functional renormalization group approach to quantum wires out of equilibrium

The functional renormalization group (FRG) provides a flexible tool to study correlations in low-dimensional electronic systems. In this paper, we present a novel FRG approach to the steady-state of quantum wires out of thermal equilibrium. Our method is correct up to second order in the two-particle interaction and accounts for inelastic scattering. We combine semi-analytic solutions of the flow equations with MPI parallelization techniques, which allows us to treat systems of up to 60 lattice sites. The equilibrium limit is well-understood and serves as a benchmark. We compute effective distribution functions, the local density of states, and the steady-state current and demonstrate that all of these quantities depend strongly on the choice of the cutoff employed within the FRG. Non-equilibrium is plagued by the lack of physical arguments in favor of a certain cutoff as well as by the appearance of secular higher-order terms which are only partly included in our approach. This demonstrates the inadequacy of a straightforward second-order FRG scheme to study interacting quantum wires out of equilibrium in the absence of a natural cutoff choice.

The Logic of Sentiment: Stowe, Hawthorne, and Melville

The Logic of Sentiment: Stowe, Hawthorne, and Melville

Examining instabilities due to driven scalars in AdS

We extend the study of the non-linear perturbative theory of weakly turbulent energy cascades in AdSd+1 to include solutions of driven systems, i.e. those with time-dependent sources on the AdS boundary. This necessitates the activation of non-normalizable modes in the linear solution for the massive bulk scalar field, which couple to the metric and normalizable scalar modes. We determine analytic expressions for secular terms in the renormalization flow equations mass values {m}_{BF}^2<{m}^2le 0 , and for various driving functions. Finally, we numerically evaluate these sources for d = 4 and discuss what role these driven solutions play in the perturbative stability of AdS.

ON TRANSVERSAL VIBRATIONS OF AN AXIALLY MOVING BEAM UNDER INFLUENCE OF VISCOUS DAMPING

In this paper, a transversal vibration of an axially moving beam under the influence of viscous damping has been studied. The axial velocity of the beam is assumed to be positive, constant and small compared to wave-velocity. The beam is moving in a positive horizontal direction between the pair of pulleys and the length between the two pulleys is fixed. From a physical viewpoint, this model describes externally damped transversal motion for a conveyor belt system. The beam is assumed to be externally damped, where there is no restriction on the damping parameter which can be sufficiently large in contrast to much research material. The straightforward expansion method is applied to obtain approximated analytic solutions. It has been shown that the obtained solutions have not been broken out for any parametric values of the small parameter 𝜀. The constructed solutions are uniform and have been damped out. Even though there are several secular terms in the solutions, but they are small compared to damping.

The Tidal Clock Effects of the Lunisolar Gravitational Field and the Earth’s Tidal Deformation

Abstract State-of-the-art atomic and optical clocks have the great potential to precisely test fundamental physical assumptions and enhance our understanding of nature. Their widespread applications require us to rigorously deduce the relativistic frequency shift in the framework of general relativity. One interesting question for clocks is tidal field effect which contains variously periodic variations. By introducing tidally deformed Earth, the general algorithms for clock and frequency comparisons are derived. We investigate the effects of external gravitational bodies and Earth’s tidal deformation on the ground- and space-based clocks. The orbital elements of satellite and locations of laboratory are introduced for corresponding clocks, in which the secular and long-period terms and short-period terms are subsequently presented. These effects called tidal clock effects produce the non-negligible contributions in the modern clock experiments and can be directly evaluated from our parameterized formulas. In addition, we also demonstrate for tidal clock effects the position and distance dependences, as well as periodicity. These tidal effects and dependences can provide valuable information for the clock comparison experiments.