Second-order Delay Differential Equations Research Articles

Oscillation by impulses for a second-order delay differential equation

We consider a certain second-order nonlinear delay differential equation and prove that the all solutions oscillate when proper impulse controls are imposed. An example is given.

Some new oscillation criteria for second order quasi-linear neutral delay differential equations

In this paper, we will establish some new oscillation criteria for second order quasi-linear neutral delay differential equations of the form [ r ( t ) | ( x ( t ) + p ( t ) x ( t - τ ) ) ′ | α - 1 ( x ( t ) + p ( t ) x ( t - τ ) ) ′ ] ′ + q ( t ) f ( x ( σ ( t ) ) ) = 0 . The results obtained essentially improve and extend those of Dzurina and Stavroulakis [J. Dzurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2003) 445–453], Sun and Meng [Y.G. Sun, F.W. Meng, Note on the paper of Dzurina and Stavroulakis, Appl. Math. Comput. 174 (2006) 1634–1641].

Stability and bifurcation of numerical discretization of a second-order delay differential equation with negative feedback

The paper analyzes a discrete second-order, nonlinear delay differential equation with negative feedback. The characteristic equation of linear stability is solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The existence of local Hopf bifurcations is investigated, and the direction and stability of periodic solutions bifurcating from the Hopf bifurcation of the discrete model are determined by the Hopf bifurcation theory of discrete system. Finally, some numerical simulations are performed to illustrate the analytical results found.

Note on the paper of Džurina and Stavroulakis

In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the form ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u [ τ ( t ) ] | α - 1 u [ τ ( t ) ] = 0 . The results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper.

GENERALIZED GALILEI-INVARIANT CLASSICAL MECHANICS

To describe the "slow" motions of n interacting mass points, we give the most general four-dimensional (4D) noninstantaneous, nonparticle symmetric Galilei-invariant variational principle. It involves two-body invariants constructed from particle 4-positions and 4-velocities of the proper orthochronous inhomogeneous Galilei group. The resulting 4D equations of motion and multiple-time conserved quantities involve integrals over the worldlines of the other n-1 interacting particles. For a particular time-asymmetric retarded (advanced) interaction, we show the vanishing of all integrals over worldlines in the ten standard 4D multiple-time conserved quantities, thus yielding a Newtonian-like initial value problem. This interaction gives 3D noninstantaneous, nonparticle symmetric, coupled nonlinear second-order delay-differential equations of motion that involve only algebraic combinations of nonsimultaneous particle positions, velocities, and accelerations. The ten 3D noninstantaneous, nonparticle symmetric conserved quantities involve only algebraic combinations of nonsimultaneous particle positions and velocities. A two-body example with a generalized Newtonian gravity is provided. We suggest that this formalism might be useful as an alternative slow-motion mechanics for astrophysical applications.

Oscillation of second-order delay differential equations on time scales

By means of Riccati transformation technique, we establish some new oscillation criteria for a second-order delay differential equation on time scales in terms of the coefficients.

Existence of positive solutions for boundary-value problems of second-order delay differential equations

The paper is concerned with the existence of positive solutions for the nonlinear eigenvalue problem with singularity and the superlinear semipositone problem of second-order delay differential equations. The main results in this paper extend some of the existing literature. The proofs are based on a fixed-point theorem in cones.

Eigenvalue criteria for existence of multiple positive solutions to boundary value problems of second-order delay differential equations

Criteria are established for existence of multiple positive solutions to the boundary value problems expressed by second-order delay differential equations of the form { − ( p ( t ) y ′ ( t ) ) ′ + q ( t ) y ( t ) = f ( t , y ( t − τ ) , y ( t + a ) ) , t ∈ ( 0 , 1 ) , α y ( t ) − β y ′ ( t ) = η ( t ) , t ∈ [ − τ , 0 ] , γ y ( t ) + δ y ′ ( t ) = ξ ( t ) , t ∈ [ 1 , 1 + a ] , where 0 < τ , a < 1 / 4 are suitably small, p ∈ C 1 [ 0 , 1 ] , p > 0 , q ∈ C [ 0 , 1 ] , and q ⩾ 0 .

Asymptotics for the Characteristic Roots of Delayed Dynamic Systems

Delayed dynamical systems appear in many areas of science and engineering. Analysis of general nonlinear delayed systems often begins with the linearized delay differential equation (DDE). The study of these linearized constant coefficient DDEs involves transcendental characteristic equations, which have infinitely many complex roots not obtainable in closed form. Here, after motivating our study with a well-known delayed dynamical system model for tool vibrations in metal cutting, we obtain asymptotic expressions for the large characteristic roots of several delayed systems. These include first- and second-order DDEs with single delays, and a first-order DDE with distributed as well as multiple incommensurate delays. For reasonable magnitudes of the coefficients of the DDEs, the approximations in each case are very good. Subsequently, a fourth delayed system involving coefficients of disparate magnitude is analyzed using an alternative asymptotic strategy. Finally, the large root asymptotics are complemented with calculations using Padé approximants to find all the roots of these systems.

Stability criteria for certain second-order delay differential equations with mixed coefficients

In this paper, we study asymptotic stability of the zero solution of the second-order linear delay differential equation: y″(t)=p 1y′(t)+p 2y′(t−τ)+q 1y(t)+q 2y(t−τ), where p 1, p 2, q 1, and q 2 are constants with p 1 p 2⩾0 and q 1 q 2<0. Here τ>0 is a constant. In proving our results we make use of Pontryagin's theory of quasi-polynomials.

Impulsive stabilization of two kinds of second-order linear delay differential equations

In this paper, we consider the impulsive stabilization problems for two kinds of 2nd-order linear delay differential equations. By the method of Lyapunov functionals, criteria on the exponential stabilization by impulses and exponential stabilization by periodical impulses are gained. The control effects of impulses are especially stressed in the definitions and results. Some examples are also given to show the applications of our results.

Oscillation of second order quasilinear delay differential equations with impulses

Sufficient conditions for oscillation of all solutions of a class of second-order quasilinear delay differential equations with fixed moments of impulse effect are found.

Remarks on the Perturbation Methods in Solving the Second-Order Delay Differential Equations

The paper presents a study on the validity of perturbation methods, suchas the method of multiple scales, the Lindstedt–Poincare method and soon, in seeking for the periodic motions of the delayed dynamic systemsthrough an example of a Duffing oscillator with delayed velocityfeedback. An important observation in the paper is that the method ofmultiple scales, which has been widely used in nonlinear dynamics, worksonly for the approximate solutions of the first two orders, and givesrise to a paradox for the third-order approximate solutions of delaydifferential equations. The same problem appears when theLindstedt–Poincare method is implemented to find the third-orderapproximation of periodic solutions for delay differential equations,though it is effective in seeking for any order approximation ofperiodic solutions for nonlinear ordinary differential equations. Apossible explanation to the paradox is given by the results obtained byusing the method of harmonic balance. The paper also indicates thatthese perturbation methods, despite of some shortcomings, are stilleffective in analyzing the dynamics of a delayed dynamic system sincethe approximate solutions of the first two orders already enable one togain an insight into the primary dynamics of the system.

On the stability of the second-order delay differential equations with variable coefficients

The sufficient conditions for the stability of the solutions of the second-order linear delay differential equations with variable coefficients is constructed.

Oscillation of a class of second-order delay differential equation with impulses

In this paper, oscillation of a class of second-order linear delay differential equation (DDE) with impulses is investigated, sufficient conditions of oscillation for all solutions of DDE with impulses are obtained by using differential inequality.

Bounded oscillation for second-order delay differential equations with unstable type in a critical case

Sharp sufficient conditions are obtained for the bounded oscillation for second-order delay differential equation with unstable type ξ″(t)=p(t)ξ(t−τ) in the critical state lim t→∞ p(t)= 2 τe 2

Oscillation criteria for second-order delay differential equations

The aim of this paper is to establish some new oscillation criteria for the second-order retarded differential equation (r(t)|u ′(t)| α−1u ′(t)) ′+p(t)|u[τ(t)]| α−1u[τ(t)]=0. The results obtained essentially improve known results in the literature.

Multiple positive solutions for boundary value problems of second-order delay differential equations

We use a fixed-point index theorem in cones to study the existence of multiple positive solutions for boundary value problems of second-order delay differential equations with the form y″(x) + ƒ(x,y(x − τ)) = 0, 0 < x < 1, y(x) = 0, −τ ≤ x ≤ 0, y(1) = 0, where 0 < τ < 1 4 is suitably small. The main result here is the generalization of Liu and Li [1] for ordinary differential equations.

Preservation of nonoscillatory behavior of solutions of second-order delay differential equations under impulsive perturbations

We offer new criteria for the preservation of nonoscillatory behavior of solutions of the delay differential equation of second-order ▪ under impulsive perturbations. A technique of direct analysis in this paper is developed.

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.