Second-order Delay Differential Equations Research Articles

Convergence of Variational Iteration Method for Second-Order Delay Differential Equations

This paper employs the variational iteration method to obtain analytical solutions of second-order delay differential equations. The corresponding convergence results are obtained, and an effective technique for choosing a reasonable Lagrange multiplier is designed in the solving process. Moreover, some illustrative examples are given to show the efficiency of this method.

Solving Second-Order Delay Differential Equations by Direct Adams-Moulton Method

This paper will consider the implementation of fifth-order direct method in the form of Adams-Moulton method for solving directly second-order delay differential equations (DDEs). The proposed direct method approximates the solutions using constant step size. The delay differential equations will be treated in their original forms without being reduced to systems of first-order ordinary differential equations (ODEs). Numerical results are presented to show that the proposed direct method is suitable for solving second-order delay differential equations.

Bifurcation of the periodic motion in nonlinear delayed oscillators

We use the residue harmonic balance scheme to study the periodic motions of a class of second-order delay-differential equations with cubic nonlinearities near and after Hopf bifurcation. The multiple solutions are found by homotopy continuation. Then, the approximation to any desired accuracy for a specific solution is captured by solving linear equations iteratively. The second-order solutions give good predictions for the frequency and amplitude, which are verified by the Runge–Kutta numerical solutions. Two typical examples, the temporal dynamics of the delay Liénard oscillator and the delay feedback Duffing system, are studied and compared. The results show how to trace analytically the relevant effect of the stiffness coefficient and the time delay on the dynamics and on the number of periodic solutions, even for large values of the bifurcation parameters.

Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators

The effects of delayed feedback terms on nonlinear oscillators have been extensively studied, and there are important applications in many areas of science and engineering. We study a particular class of second-order delay-differential equations near a point of triple-zero nilpotent bifurcation. Using center manifold and normal form reduction, we show that the three-dimensional nonlinear normal form for the triple-zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay-differential equation.

Stability and bifurcation analysis in the delay-coupled nonlinear oscillators

This paper investigates the dynamical behavior of two oscillators with nonlinearity terms, which are coupled with finite delay parameters. Each oscillator is a general class of second-order nonlinear delay-differential equations. The system of delay differential equations is analyzed by reducing the delay equations to a system of ordinary differential equations on a finite-dimensional center manifold, the corresponding to an infinite-dimensional phase space. In addition, the characteristic equation for the linear stability of the trivial equilibrium is completely analyzed and the stability region is illustrated in the parameters space. Our analysis reveals necessary coefficients of the reduced vector field on the center manifold for studying the bifurcations of the trivial equilibrium such as transcritical, pitchfork, and Hopf bifurcation. Finally, we consider the delay-coupled van der Pol equations.

Asymptotic behavior of solutions of nonlinear delay differential equations with impulse

This paper studies the asymptotic behavior of solutions of the second-order nonlinear delay differential equations with impulses: and some sufficient conditions are obtained. Key words: Asymptotic behavior, second-order nonlinear delay differential equation, impulses.

ON THE OSCILLATION OF SECOND-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES

By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criterion for the second-order nonlinear delay dynamic equations on a time scale , here is the ratio of two positive odd integers with and real-valued positive right-dense continuous functions defined on . Our results not only extend and improve some known results, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.

Oscillation theorems for second-order nonlinear neutral differential equations

The purpose of this paper is to study the oscillation of the second-order neutral differential equations of the form ( a ( t ) [ z ′ ( t ) ] γ ) ′ + q ( t ) x β ( σ ( t ) ) = 0 , where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) . We explore properties of given equations by examining those of associated first-order delay equations. New comparison theorems essentially simplify the examination of the equations studied as they allow us to deduce the oscillation of the second-order delay differential equation by applying the oscillation criteria obtained to the first-order delay equations. The results obtained are easy to verify.

Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales (Ⅱ)

By using the generalized Riccati transformation and the inequality technique,we establish four new oscillation criteria for certain second-order half-linear delay dynamic equations with damping on time scales.Our results extend and improve some known results,but also unify the oscillation of the second-order half-linear delay differential equations with damping and the second-order half-linear delay difference equations with damping.

Fitted finite difference method for singularly perturbed delay differential equations

This paper deals with singularly perturbed initial value problem for linear second-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.

Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid

Matrix methods, now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineers for which the existing methodologies do not give reliable results, these methods are solving them competitively. In this work, a matrix methods is presented for approximate solution of the second-order singularly-perturbed delay differential equations. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The error analysis and convergence for the proposed method is introduced. Finally some experiments and their numerical solutions are given.

Positive solutions of second-order delay differential equations with a damping term

In the paper, the existence of positive solutions is studied for the second-order delay differential equation with a damping term ẍ(t)+a(t)ẋ(t)+b(t)x(h(t))=0 using a comparison with the integro-differential equation ẏ(t)+∫t0te−∫sta(ξ)dξb(s)y(h(s))ds=0. Explicit non-oscillation criteria and comparison type results are derived.

A uniform numerical method for dealing with a singularly perturbed delay initial value problem

This work deals with a singularly perturbed initial value problem for a quasi-linear second-order delay differential equation. An exponentially fitted difference scheme is constructed, in an equidistant mesh, which gives first-order uniform convergence in the discrete maximum norm. Numerical results are also presented.

Strongly Nonlinear Oscillators Subject to Delay

A large class of strongly nonlinear conservative oscillators subject to a delayed feedback are modeled mathematically by second-order delay differential equations. Recent applications include the control of crane oscillations and lasers subject to optoelectronic feedback. We apply the method of averaging in the case of weak damping and weak feedback and determine the bifurcation diagram of the limit-cycle solutions. We find that the coexistence of a stable equilibrium with one or several stable periodic solutions is unavoidable if the delay is sufficiently large.

High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations

This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical harmonic balance method are highlighted and discussed.

Stability of Delay Equations Written as State Space Models

In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

Numerical method for a singularly perturbed convection–diffusion problem with delay

This paper deals with the singularly perturbed boundary value problem for a linear second-order delay differential equation. For the numerical solution of this problem, we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that one gets first order convergence in the discrete maximum norm, independently of the perturbation parameter. Numerical results are presented which illustrate the theoretical results.

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

Some sufficient conditions are established for the oscillation of second-order neutral differential equation , , where . The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 (1985), Grace and Lalli (1987), Ruan (1993), H. J. Li (1996), H. J. Li (1997), Xu and Xia (2008).

On Connection between Second-Order Delay Differential Equations and Integrodifferential Equations with Delay

On Connection between Second-Order Delay Differential Equations and Integrodifferential Equations with Delay

Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales

By means of the averaging technique and the generalized Riccati transformation technique, we establish some oscillation criteria for the second-order quasilinear neutral delay dynamic equations , , where , and the time scale interval is . Our results in this paper not only extend the results given by Agarwal et al. (2005) but also unify the oscillation of the second-order neutral delay differential equations and the second-order neutral delay difference equations.