Nonlinear Delay Differential Equations Research Articles

On the oscillation of nonlinear delay differential equations and their applications

Abstract The oscillation of nonlinear differential equations is used in many applications of mathematical physics, biological and medical physics, engineering, aviation, complex networks, sociophysics and econophysics. The goal of this study is to create some new oscillation criteria for fourth-order differential equations with delay and advanced terms ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + ∑ j = 1 r β j ( x ) w k ( γ j ( x ) ) = 0 , {({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+\mathop{\sum }\limits_{j=1}^{r}{\beta }_{j}(x){w}^{k}({\gamma }_{j}(x))=0, and ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + a 2 ( x ) h ( w ‴ ( x ) ) + β ( x ) f ( w ( γ ( x ) ) ) = 0 . {({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+{a}_{2}(x)h({w}^{\prime\prime\prime }(x))+\beta (x)f(w(\gamma (x)))=0. The method is based on the use of the comparison technique and Riccati method to obtain these criteria. These conditions complement and extend some of the results published on this topic. Two examples are provided to prove the efficiency of the main results.

On meromorphic solutions of nonlinear delay-differential equations

Using Cartan's second main theorem and Nevanlinna's theorem concerning a group of meromorphic functions, we obtain the growth and zero distribution of meromorphic solutions of the nonlinear delay-differential equation fn(z)+P(z)f(k)(z+η)=H0(z)+H1(z)eω1zq+⋯+Hm(z)eωmzq, where n,k,q,m are positive integers, η,ω1,⋯,ωm are complex numbers with ω1⋯ωm≠0, and P,H0,H1,⋯,Hm are entire functions of order less than q with PH1⋯Hm≢0. Especially for η=0, some sufficient conditions are given to guarantee the above equation has no meromorphic solutions of few poles.

Investigation of nonlinear fractional delay differential equation via singular fractional operator

Abstract The present research work is basically devoted to construction of a fractional order differential equation with time delay. Initially, integral representation is given to solution of the underline problem. Afterwards, operator form of solution is studied under some auxiliary hypothesis. Since uniqueness of solution is required, therefore we also provide results for exploring the uniqueness of solution for the underlying model. Using Lebesgue dominated convergence theorem and some other results from analysis, this work provides results devoted to existence of at least one solution. Also, for investigating the nature of solution for the proposed model, we study different kind of stability analysis. These stability related results show, how the solution behave with time. At the end of the article, we illustrate the obtained results via some examples.

An a priori error analysis of the hp-version of the C 0-continuous Petrov-Galerkin method for nonlinear second-order delay differential equations

We study an hp-version of the -continuous Petrov-Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bound in the -norm that is fully explicit in the local time steps, in the local approximation degrees, and in the local regularity indexes of the exact solutions. Moreover, we prove that the -continuous Petrov-Galerkin based on special hp-version discretization can yield exponential rates of convergence for analytic solutions with initial singularities. Numerical experiments are provided to illustrate the theoretical results.

Optimization with delay-induced bifurcations.

Optimization is finding the best solution, which mathematically amounts to locating the global minimum of some cost function. Optimization is traditionally automated with digital or quantum computers, each having their limitations and none guaranteeing an optimal solution. Here, we conceive a principle behind optimization based on delay-induced bifurcations, which is potentially implementable in non-quantum analog devices. Often, optimization techniques are interpreted via a particle moving in multi-well energy landscape and to prevent confinement to a non-global minima they should incorporate mechanisms to overcome barriers between the minima. Particularly, simulated annealing digitally emulates pushing a fictitious particle over a barrier by random noise, whereas quantum computers utilize tunneling through barriers. In our principle, the barriers are effectively destroyed by delay-induced bifurcations. Although bifurcation scenarios in nonlinear delay-differential equations can be very complex and are notoriously difficult to predict, we hypothesize, verify, and utilize the finding that they become considerably more predictable in dynamical systems, where the right-hand side depends only on the delayed variable and represents a gradient of some potential energy function. By tuning the delay introduced into the gradient descent setting, thanks to global bifurcations destroying local attractors, one could force the system to spontaneously wander around all minima. This would be similar to noise-induced behavior in simulated annealing but achieved deterministically. Ideally, a slow increase and then decrease of the delay should automatically push the system toward the global minimum. We explore the possibility of this scenario and formulate some prerequisites.

Chatter in milling with robots with structural nonlinearity

Stiffness and damping nonlinearities are often negligible in machine tool vibrations, but they are shown to be prominent in industrial robots that are used for milling, mainly due to the several nonlinear phenomenon that occur in the robot’s revolute joints. In this paper, a 2DOF nonlinear vibratory model is presented to study regenerative chatter in robotic milling. The presented model includes cubic stiffness and damping terms to account for the robot’s structural nonlinearities. The machining forces are represented by a combination of periodic forces and delayed vibration terms; the former is generated by the feed motion of the tool and the later by chip regeneration. Furthermore, the machining force model is non-smooth due to (a) periodic entrance and exit of the cutting edges to and from the cutting region, and (b) loss-of-contact at high-amplitude oscillations. The dynamics in the presented model is therefore described by a set of nonlinear Delay Differential Equations with periodic coefficients and non-smooth forces. Numerical continuation with smoothing techniques is used to determine the combinations of spindle speed and axial depth of cut at which the forced periodic vibrations go through secondary Hopf or period doubling bifurcations, causing chatter in the milling process. A KUKA robotic arm with milling end-effector is used as a case study. The presented analysis shows that feedrate, which has no effect on chatter in machine tools with linear dynamics, alters the stability of vibrations in milling systems with nonlinear structural dynamics. Nonlinear systems’ stiffness and damping depend on vibration amplitude. Since feedrate directly influences the amplitude of forced vibrations, the nonlinear system’s damping and stiffness and consequently its chatter stability changes by feedrate.

Oscillation and Asymptotic Properties of Differential Equations of Third-Order

The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator (a(ι)[(b(ι)x(ι)+p(ι)x(ι−τ)′)′]β)′+∫cdq(ι,μ)xβ(σ(ι,μ))dμ=0, where ι≥ι0 and w(ι):=x(ι)+p(ι)x(ι−τ). New oscillation results are established by using the generalized Riccati technique under the assumption of ∫ι0ιa−1/β(s)ds<∫ι0ι1b(s)ds=∞asι→∞. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.

Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

Numerical treatment of a fractional order system of nonlinear stochastic delay differential equations using a computational scheme

In this article, a step-by-step collocation approach based on the shifted Legendre polynomials is presented to solve a fractional order system of nonlinear stochastic differential equations involving a constant delay. The problem is considered with suitable initial condition and the fractional derivative is in the Caputo sense. With a step-by-step process, first, the considered problem is converted into a non-delay fractional order system of nonlinear stochastic differential equations in each step and then, a shifted Legendre collocation scheme is introduced to solve this system. By collocating the obtained residual at the shifted Legendre points, we get a nonlinear system of equations in each step. The convergence analysis and rate of convergence of the proposed method are investigated . Finally, three test examples are provided to affirm the accuracy of this technique in the presence of different noise measures.

On the Oscillation of second order nonlinear neutral delay differential equations

AbstractA class of second-order neutral delay differential equations is considered. New oscillation criteria are established to complement and improve some known results in the literature. Two examples are given to support our results.

Analysis of positive fractional-order neutral time-delay systems

In this paper, we consider an initial value problem for linear matrix coefficient systems of the fractional-order neutral differential equations with two incommensurate constant delays in Caputo’s sense. Firstly, we introduce the exact analytical representation of solutions to linear homogeneous and non-homogeneous neutral fractional-order differential-difference equations system by means of newly defined delayed Mittag–Leffler type matrix functions. Secondly, a criterion on the positivity of a class of fractional-order linear homogeneous time-delay systems has been proposed. Furthermore, we prove the global existence and uniqueness of solutions to non-linear fractional neutral delay differential equations system using the contraction mapping principle in a weighted space of continuous functions with regard to classical Mittag–Leffler functions. In addition, Ulam–Hyers stability results of solutions are attained based on fixed-point approach. Finally, we present an example to demonstrate the applicability of our theoretical results.

Mean-square stability of the zero equilibrium of the nonlinear delay differential equation: Nicholson’s blowflies application

We are concerned about the stochastic nonlinear delay differential equation. The stochasticity arises from the white Gaussian noise, which is the time derivative of the standard Brownian motion. The main objective of this paper is to introduce a new technique using the Lyapunov functional for the study of stability of the zero solution of the stochastic delay differential system. Constructing a new appropriate deterministic system in the neighborhood of the origin is an effective way to investigate the necessary and sufficient conditions of stability in the sense of the mean square. Nicholson’s blowflies equation is one of the major problems in ecology; necessary conditions for the possible extinction of the Nicholson’s blowflies population are investigated. We support our theoretical results by providing areas of stability and some numerical simulations of the solution of the system using the Euler–Maruyama scheme, which is mean square stable Maruyama (Rendiconti del Circolo Matematico di Palermo 4(1):48, 1955), Cao et al. (Appl Math Comput 159(1):127–135, 2004).

Fixed solutions of second order nonlinear difference equations

The paper is concerned with asymptotic and stability behaviors of fixed solutions of the second order nonlinear delay difference equation of the form ?2(xn +pnxn-k -qnxn-l)+ f(xn) = 0, n = 0,1,2, · ··,. Examples are provided to illustrate the results.

The oscillation of lasota-wazewska model with a variable probability of death of red blood cell

In this paper, the Lasota-Wazewska model of survival of red blood cell in humans was studied, in which the probability of death of red blood cells was adopted as a function using nonlinear delay differential equation of the first order. Some conditions were established to guarantee that the number of red blood cells oscillates about the equilibrium. The results are supported by some illustrative examples.

Modeling of chatter vibrations in gun drilling process

The dynamics of the gun drilling process is analyzed in this paper. The tool shank is modeled as long straight beam vibrating in transverse direction under action of cutting forces. Axial force component is expressed as proportional to cutting thickness, which is determined as nonlinear function of beam transverse deflection with time delay. Nonlinear equations of motion of the drilling shank are derived. The stability diagram of the system dynamics was determined. The bifurcation analysis of nonlinear differential delay equations by means of multiple scale method was performed. The obtained results were verified by numerical integration of nonlinear equations. The influence of cutting conditions on system stability and chatter amplitude was observed.

An iterative scheme for the oscillation criteria of a nonlinear delay differential equation with several deviating arguments

This paper deals with the oscillatory results of first order nonlinear delay differential equations with several deviating arguments by employing an iterative process. The results presented here has improved the outcomes of [1, 2, 8]. Various examples are solved in MATLAB software to illustrate the relevance of the main results.

The strong convergence and stability of explicit approximations for nonlinear stochastic delay differential equations

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under less restrictive conditions, the truncated Euler-Maruyama (TEM) schemes for SDDEs are proposed, which numerical solutions are bounded in the q th moment for q ≥ 2 and converge to the exact solutions strongly in any finite interval. The 1/2 order convergence rate is yielded. Furthermore, the long-time asymptotic behaviors of numerical solutions, such as stability in mean square and $\mathbb {P}-1$ , are examined. Several numerical experiments are carried out to illustrate our results.

A new accurate procedure for solving nonlinear delay differential equations

In this paper, a modified algorithm based on the residual power series procedure is employed to find an accurate approximate solution for nonlinear delay differential equations (NDDEs). This modification is considered as a powerful procedure for improving the efficiency of the residual power series method (RPSM) by using the Laplace transform and Pade approximant to be an effective procedure that has the ability to give accurate results ´ closed to the exact solutions with easy computational work. Some numerical examples are presented to check the validity and the applicability of this modification and the results obtained are compared by the results obtained by other method in literature to illustrate and prove it is efficiency and reliability for solving this kind of equations.

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix

Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix