Asymptotic Oscillation Research Articles

Asymptotic Properties and Oscillation of the Solutions in a Periodic Impulsive Hematopoiesis Model

Asymptotic Properties and Oscillation of the Solutions in a Periodic Impulsive Hematopoiesis Model

The asymptotic behavior and oscillation for a class of third-order nonlinear delay dynamic equations

The asymptotic behavior and oscillation for a class of third-order nonlinear delay dynamic equations

On the Global Asymptotic Stability and 4-Period Oscillation of the Third-Order Difference Equation

The main objective of this paper is to study the global behavior and oscillation of the following third-order rational difference equation x n + 1 = α x n x n − 1 x n − 2 / β x n − 1 2 + γ x n − 2 2 , where the initial conditions x − 2 , x − 1 , x 0 are nonzero real numbers and α , β , γ are positive constants such that α ≤ β + γ . Visual examples supporting solutions are given at the end of the study. The figures are found with the help of MATLAB.

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

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.

Differential equations of the neutral delay type: More efficient conditions for oscillation

<abstract><p>In this article, we derive an optimized relationship between the solution and its corresponding function for second- and fourth-order neutral differential equations (NDE) in the canonical case. Using this relationship, we obtain new monotonic properties of the second-order equation. The significance of this paper stems from the fact that the asymptotic behavior and oscillation of solutions to NDEs are substantially affected by monotonic features. Based on the new relationships and properties, we obtain oscillation criteria for the studied equations. Finally, we present examples and review some previous theorems in the literature to compare our results with them.</p></abstract>

Properties of the ionisation glitch

Context. Determining the properties of solar-like oscillating stars can be subject to many biases. A particularly important example is the helium-mass degeneracy, where the uncertainties regarding the internal physics can cause a poor determination of both the mass and surface helium content. Accordingly, an independent helium estimate is needed to overcome this degeneracy. A promising way to obtain such an estimate is to exploit the so-called ionisation glitch, that is, the deviation from the asymptotic oscillation frequency pattern caused by the rapid structural variation in the He ionisation zones. Aims. Although it is progressively becoming more sophisticated, the glitch-based approach faces problems inherent to its current modelling such as the need for calibration using realistic stellar models. This requires a physical model of the ionisation region that explicitly involves the parameters of interest, such as the surface helium abundance, Ys. Methods. Through a thermodynamic treatment of the ionisation region, an analytical approximation for the first adiabatic exponent Γ1 is presented. Results. The induced stellar structure is found to depend on only three parameters, including the surface helium abundance Ys and the electron degeneracy ψCZ in the convective region. The model thus defined allows a wide variety of structures to be described, and it is in particular able to approximate a realistic model in the ionisation region. The modelling work we conducted enables us to study the structural perturbations causing the glitch. More elaborate forms of perturbations than those that are usually assumed are found. It is also suggested that there might be a stronger dependence of the structure on the electron degeneracy in the convection zone and on the position of the ionisation region rather than on the amount of helium itself. Conclusions. When analysing the ionisation glitch signature, we emphasise the importance of having a relation that can take these additional dependences into account.

Oscillatory Properties of Odd-Order Delay Differential Equations with Distribution Deviating Arguments

Throughout this work, new criteria for the asymptotic behavior and oscillation of a class of odd-order delay differential equations with distributed deviating arguments are established. Our method is essentially based on establishing sharper estimates for positive solutions of the studied equation, using an iterative technique. Moreover, the iterative technique allows us to test the oscillation, even when the related results fail to apply. By establishing new comparison theorems that compare the nth-order equations with one or a couple of first-order delay differential equations, we obtain new conditions for oscillation of all solutions of the studied equation. To show the importance of our results, we provide two examples.

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

The aim of this study was to examine the asymptotic properties and oscillation of the even-order neutral differential equations. The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order delay equations. Our results improve and complement some well-known results. We obtain Hille and Nehari type oscillation criteria to ensure the oscillation of the solutions of the equation. One example is provided to illustrate these results.

Localized mode and nonergodicity of a harmonic oscillator chain

We present a simple and microscopic physical model that breaks the ergodicity. Our model consists of coupled classical harmonic oscillators, and the motion of the tagged particle obeys the generalized Langevin equation satisfying the second fluctuation dissipation theorem. It is found that although the nonergodicity strength, which is expected to detect the ergodicity breaking, for this model vanishes, the velocity auto correlation function of the tagged particle asymptotically oscillates. We analyze the model by using the molecular dynamics and the exact diagonalization as well as the rigorous mapping to the generalized Langevin equation. Our analysis reveals that the asymptotic oscillation is caused by a localized mode with an isolated frequency from the continuous phonon spectrum.

BEHAVIOR OF POSITIVE SOLUTIONS OF A DIFFERENCE EQUATION

In this paper we deal with the difference equation <TEX>$$y_{n+1}=\frac{ay_{n-1}}{by_ny_{n-1}+cy_{n-1}y_{n-2}+d}$$</TEX>, <TEX>$$n{\in}\mathbb{N}_0$$</TEX>, where the coefficients a, b, c, d are positive real numbers and the initial conditions <TEX>$y_{-2}$</TEX>, <TEX>$y_{-1}$</TEX>, <TEX>$y_0$</TEX> are nonnegative real numbers. Here, we investigate global asymptotic stability, periodicity, boundedness and oscillation of positive solutions of the above equation.

On the global asymptotic stability and oscillation of solutions in a stochastic business cycle model

We discuss a second order nonlinear stochastic difference equation which is constructed of a business cycle model with organized labor considered. A global asymptotic mean square stability criterion is obtained by Lyapunov function method. We also prove a theorem on the almost sure oscillation of the solutions for the difference equation with state-independent stochastic perturbations.

On the oscillation of odd order advanced differential equations

The aim of this paper is to study the asymptotic properties and oscillation of the nth order advanced differential equations (r(t) x (n–1) (t) γ ) � + q(t)x γ τ (t) =0 . The results obtained are based on the Riccati transformation. MSC: 34K11; 34C10

A review on numerical schemes for solving a linear stochastic oscillator

In recent years several numerical methods to solve a linear stochastic oscillator with one additive noise have been proposed. The usual aim of these approaches was to preserve different long time properties of the oscillator solution. In this work we collect these properties, namely, symplecticity, linear growth of its second moment and asymptotic oscillation around zero. We show that these features can be studied in terms of the coefficients of the matrices that appear in the linear recurrence obtained when the schemes are applied to the oscillator. We use this study to compare the numerical schemes as well as to propose new schemes improving some properties of classical methods.

Oscillation criteria and asymptotic properties of solutions of third‐order nonlinear neutral differential equations

The main goal of this article is to study the asymptotic properties and oscillation of the third‐order neutral differential equations with discrete and distributed delay. We give several theorems and related examples to illustrate the applicability of these theorems. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Nonlinear aerodynamic forces on thin flat plate: Numerical study

This paper first presents a numerical simulation of nonlinear aerodynamic forces on a thin flat plate through an integration of the computational fluid dynamics (CFD) method and the forced asymptotic oscillation method. The thin flat plate is forced to have either asymptotic torsional oscillation or asymptotic vertical oscillation of increasing amplitude. A multiple-domain mesh technique together with unstructured dynamic meshes is used in the CFD simulation to accommodate large amplitude oscillations of the plate. The instantaneous frequencies and amplitudes of the aerodynamic forces are then identified from the simulated asymptotic aerodynamic force time histories using the continuous wavelet transform (CWT) in terms of the CWT ridges. Extensive numerical studies are finally performed to examine the feasibility of the proposed approach. The results show that the CFD method used in this study can properly simulate nonlinear aerodynamic forces on the plate. The amplitude of the aerodynamic force depends on the amplitude of the forced oscillation and there are high-order harmonic aerodynamic forces of higher frequency than the forced oscillation frequency, both indicating the nonlinearity of aerodynamic forces. The results also show the flutter derivatives associated with self-excited aerodynamic forces depend on the amplitude of forced oscillation in addition to reduced velocity.

DYNAMICS OF A NON-AUTONOMOUS HIV-1 INFECTION MODEL WITH DELAYS

In this paper, following a previous paper ([32] Permanence and extinction of a non-autonomous HIV-1 model with two time delays, preprint) on the permanence and extinction of a delayed non-autonomous HIV-1 within-host model, we introduce and investigate a delayed HIV-1 model including maximum homeostatic proliferation rate of CD4+T-cells and varying coefficients. By applying the asymptotic analysis theory and oscillation theory, we show: (i) the system will be permanent when the threshold value R*&gt; 1, and for this case we also obtain the explicit estimate of the eventual lower bound of the HIV-1 virus load; (ii) the threshold value R* &lt; 1 implies the extinction of the virus. Furthermore, we obtain that the threshold dynamics is in agreement with that of the corresponding autonomous system, which extends the classic results for the system with constant coefficients. Numerical simulations are also given to illustrate our main results, and in particular, some sensitivity test of R*is established.

Dynamics of a discontinuous discrete Beverton–Holt model

Our aim is to investigate the global asymptotic behaviour, oscillation, periodicity and bifurcation in discontinuous Beverton–Holt type difference equationwhere x0>0, functions k and r are discontinuous piecewise constant functionssatisfyingand h is Heaviside function.

On the oscillation of higher-order delay differential equations

The aim of this paper is to study the asymptotic properties and oscillation of the nth-order delay differential equation E $$ {{\left( {r(t){{{\left[ {{z^{{\left( {n-1} \right)}}}(t)} \right]}}^{\gamma }}} \right)}^{\prime }}+q(t)f\left( {x\left( {\tau (t)} \right)} \right)=0. $$ The results obtained are based on some new comparison theorems that reduce the problem of oscillation of an nth-order equation to the problem of oscillation of one or more first-order equations. We handle both cases $$ \begin{array}{*{20}{c}} {\int\limits^{\infty } {{r^{{-1/\gamma }}}(t)dt=\infty } } & {\mathrm{and}} & {\int\limits^{\infty } {{r^{{-1/\gamma }}}(t)dt<\infty .} } \end{array} $$ The comparison principles simplify the analysis of Eq. (E).

Oscillation theorems for higher order neutral differential equations

The aim of this paper is to study the asymptotic properties and oscillation of the nth order neutral differential equations(E)r(t)x(t)+p(t)x(τ(t))(n-1)′+q(t)x(σ(t))=0.Obtained results are based on the new comparison theorems, that permit to reduce the problem of the oscillation of the nth order equation to the oscillation of a set of the first order equation. Obtained comparison principles essentially simplify the examination of studied equations and allow to relax some conditions imposed on the coefficients of (E).

Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of thenth order delay differential equations(E)(r(t)[x(n−1)(t)]γ)′+q(t)xγ(τ(t))=0. The results obtained utilize also the comparison theorems.