Methods For Delay Differential Equations Research Articles

Construction of highly stable parallel two-step Runge–Kutta methods for delay differential equations

It is shown that any A-stable two-step Runge–Kutta method of order p ˜ and stage order q = p ˜ for ordinary differential equations can be extended to the P-stable method of uniform order p = p ˜ for delay differential equations.

Computing breaking points in implicit delay differential equations

Systems of implicit delay differential equations, including state-dependent problems, neutral and differential-algebraic equations, singularly perturbed problems, and small or vanishing delays are considered. The numerical integration of such problems is very sensitive to jump discontinuities in the solution or in its derivatives (so-called breaking points). In this article we discuss a new strategy – peculiar to implicit schemes – that allows codes to detect automatically and then to compute very accurately those breaking points which have to be inserted into the mesh to guarantee the required accuracy. In particular for state-dependent delays, where breaking points are not known in advance, this treatment leads to a significant improvement in accuracy. As a theoretical result we obtain a general convergence theorem which was missing in the literature (see Bellen and Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003). Furthermore, as a useful by-product, we design strategies that are able to detect points of non-uniqueness or non-existence of the solution so that the code can terminate when such a situation occurs. A new version of the code RADAR5 together with drivers for some real-life problems is available on the homepages of the authors.

Nonlinear stability of one-leg methods for delay differential equations of neutral type

This paper is devoted to investigations into numerical stability properties of one-leg methods for nonlinear neutral delay differential equations. At first, a series of new stability concepts, such as GS- stability, GAS- stability and Weak GS- stability, are introduced. Then it is proved that a strongly A - stable one-leg method with linear interpolation is GAS- stable, and that an A - stable one-leg method with linear interpolation is GS- stable and Weakly GS- stable. Some numerical experiments are given in the last section of this paper which confirm our results.

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation d u / d t = λ u ( t ) + μ u ( t - τ ) , where τ is a constant delay and λ , μ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.

Numerical dissipativity of multistep methods for delay differential equations

Dissipative differential equations have frequently appeared in the fields of physics, engineering, and biology. In this paper we investigate numerical dissipativity of linear multistep and one-leg methods applied to a class of dissipative delay differential equations. We show that for such class of dissipative systems these numerical methods are dissipative if and only if they are A-stable for ordinary differential equations. One numerical experiment is given to illustrate our result.

Rational approximation method for delay differential equations with proportional delay

Consider the pantograph differential equation y′( t) = ay( t) + by( qt) + f( t), 0 ⩽ t ⩽ T, y(0) = y 0, with proportional delay qt, 0 < q < 1. In this paper, we propose a piecewise (2 m, m)-rational approximation with “quasi-uniform meshes” which corresponds to mth collocation method, and establish the global error analysis of O( h 2 m ) on successive mesh points. This method is useful than the known collocation method in case that a long time integration is needed, that is, if T is large, then the number of steps in our method is less than that of the collocation method. Numerical experiments are also contained.

Method of lines approximations of delay differential equations

We consider a special type of numerical methods for delay differential equations (DDEs). By introducing a new independent variable, an initial value problem for DDEs is converted into an initial-boundary value problem for the convection equation. Thus, it is also possible to get an approximate solution to the DDE problem by solving the initial-boundary value problem with a suitable numerical method instead of solving the original problem. In this paper, we study a family of method of lines (MOL) approximations to the problem, which is obtained by applying Runge-Kutta (RK) methods for space discretization, and prove their convergence under the assumption that the RK methods satisfy a condition, known as an algebraic characterization of A-stability. The result is also confirmed by numerical experiments. Moreover, we show that the condition derives several stability properties of the MOL approximations.

The numerical analysis of functional integral and integro-differential equations of Volterra type

The qualitative and quantitative analysis of numerical methods for delay differential equations is now quite well understood, as reflected in the recent monograph by Bellen and Zennaro (2003). This is in remarkable contrast to the situation in the numerical analysis of functional equations, in which delays occur in connection with memory terms described by Volterra integral operators. The complexity of the convergence and asymptotic stability analyses has its roots in new ‘dimensions’ not present in DDEs: the problems have distributed delays; kernels in the Volterra operators may be weakly singular; a second discretization step (approximation of the memory term by feasible quadrature processes) will in general be necessary before solution approximations can be computed. The purpose of this review is to introduce the reader to functional integral and integro-differential equations of Volterra type and their discretization, focusing on collocation techniques; to describe the ‘state of the art’ in the numerical analysis of such problems; and to show that - especially for many ‘classical’ equations whose analysis dates back more than 100 years - we still have a long way to go before we reach a level of insight into their discretized versions to compare with that achieved for DDEs.

Hopf bifurcation cascade in small-αlaser diodes subject to optical feedback

We analyze theoretically the dynamics of a semiconductor laser subject to optical feedback, on the basis of the well-known Lang-Kobayashi equations. Previous investigations on this laser system suggest that a small linewidth enhancement factor (alpha factor) stabilizes the laser dynamics. By contrast, we unveil here optical feedback induced instabilities which are present for a small value of alpha but which disappear when alpha increases above alpha approximately 1. By combining numerical simulations and modern continuation methods for delay-differential equations, we unveil cascades of subcritical and supercritical Hopf bifurcations on the first external-cavity mode (ECM). We unveil for the first time, to our knowledge, the occurrence of subcritical Hopf bifurcation points for intermediate values of the EC length, i.e. close to the boundary between the short and the long EC regimes. They lead to severe laser instabilities such as large intensity and possibly chaotic pulsations. Moreover, these Hopf bifurcation cascades for small values of alpha are shown to be responsible for different bifurcation scenarios leading to restabilization of the first ECM and to ECM bistability.

Book Review: Numerical Methods for Delay Differential Equations. by Alfredo Bellen and Marino Zennaro

Book Review: Numerical Methods for Delay Differential Equations. by Alfredo Bellen and Marino Zennaro

DYNAMICS OF LINEAR MULTISTEP METHODS FOR DELAY DIFFERENTIAL EQUATIONS

In this paper we study the relationship between the asymptotic behavior of a numerical simulation by linear multistep method and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of linear multistep method for nonlinear delay differential equation is investigated and we prove that A-stable linear multistep methods are NP-stable. It is shown that a consistent zero-stable linear multistep method does not admit spurious fixed points. The existence of spurious period-two solutions in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period-two solutions.

Exponential convergence estimates for neural networks with multiple delays

Componentwise estimates of (global) exponential convergence are obtained for a class of neural networks with multiple delays by com- parison method for delay differential equations. The method is simple and straightforward and does not resort to any Lyapunov functional. The main result shows explicitly the effect of time delays on exponential decay rate of the networks and is hence of practical interest in designing a fast and stable neural network. Compared with some existing results for stability of the Hopfield network model and delayed cellular neural network model via Lyapunov method, the present results can provide further detailed charac- terization of the convergence behavior for the systems under the same or similar conditions.

Convergence Aspects of Step-Parallel Iteration of Runge-Kutta Methods for Delay Differential Equations

Implicit Runge-Kutta methods are known as highly accurate and stable methods for solving differential equations. However, the iteration technique used to solve implicit Runge-Kutta methods requires a lot of computational efforts. To lessen the computational effort, one can iterate simultaneously at a number of points along the t-axis. In this paper, we extend the PDIRK (Parallel Diagonal Iterated Runge-Kutta) methods to delay differential equations (DDEs). We give the region of convergence and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector. It is proved that PDIRK methods to DDEs are efficient, and the diagonal matrix D of the PDIRK methods for DDES can be selected in the same way as for ordinary differential equations (ODEs).

Stability analysis of two-step Runge-Kutta methods for delay differential equations

We investigate stability properties of two-step Runge-Kutta methods with respect to the linear test equation y′( t) = ay( t) + by( t − τ), t ≥ 0, y( t) = g( t), t ϵ [−τ,0], where a and b are complex parameters. It is known that the solution y( t) to this equation tends to zero as t → ∞ if | b| < - Re( a). We will show that under some conditions this property is inherited by any A-stable two-step Runge-Kutta method applied on a constrained mesh to delay differential equations, i.e., that the corresponding method is P-stable.

Variable multistep methods for delay differential equations

In this paper, variable stepsize multistep methods for delay differential equations of the type y( t) = f( t, y( t), y( t − τ)) are proposed. Error bounds for the global discretization error of variable stepsize multistep methods for delay differential equations are explicitly computed. It is proved that a variable multistep method which is a perturbation of strongly stable fixed step size method is convergent.

Nonlinear Stability of General Linear Methods for Delay Differential Equations

This paper is devoted to a study of nonlinear stability of general linear methods for the numerical solution of delay differential equations in Hilbert spaces. New stability concepts are further introduced. The stability properties of (k,p,q)-algebraically stable general linear methods with piecewise constant or linear interpolation procedure are investigated. We also discuss stability of linear multistep methods viewed as a special subset of the class of general linear methods.

Regularity properties of one-leg methods for delay differential equations

The main purpose of this paper is to investigate the asymptotic states of one-leg methods for multidelay differential equations. In particular, the existence of spurious steady solutions and period-2 solutions in constant or variable timestep is studied, and the concepts of R [1]-regularity and R [2]-regularity of one-leg methods for multidelay differential equations are introduced and studied. Some conditions guaranteeing R [1]-regularity and R [2]-regularity of such methods applied to multidelay differential equations with some important structures are given.

Stability of implicit one-block methods for delay differential equations

This paper deals with the stability analysis of implicit one-block methods for the numerical solutions of the systems of delay differential equations (DDEs). We focus on the behavior of such methods when they are applied to the linear test problem y ′ (t)=Ly(t)+My(t−τ), t⩾0, y(t)=g(t), −τ⩽t⩽0 with τ>0 and L, M constant complex-value matrices. We show that an A-stable implicit one-block method preserves the asymptotic stability property of the analytical solutions of the system (0.1).

Using resolvent conditions to obtain new stability results for -methods for delay differential equations

By using the Kreiss resolvent condition we establish upper bounds for the growth of errors in two simple, but typical, numerical processes for solving delay differential equations. We show that, when the θ‐methods are used, with a suitably restricted stepsize, for solving the linear test problems Z′(t) = λZ(t) + μZ(t − τ), errors grow at most linearly with the number of timesteps and with the dimension involved. Moreover, we investigate whether this kind of error growth is valid uniformly within the so‐called stability regions of these methods. Finally some numerical experiments are carried out with a representative member of the class of processes under consideration, and it is shown that our theoretical bounds for the error growth are of practical interest.

Geometric proofs of numerical stability for delay equations

In this paper, asymptotic stability properties of implicit Runge–Kutta methods for delay differential equations are considered with respect to the test equation y′ (t) = a y (t) + b y(t − 1) with a, b ∈ ∁. In particular, we prove that symmetric methods and all methods of even order cannot be unconditionally stable with respect to the considered test equation, while many of them are stable on problems where a ∈ ℜ and b ∈ ∁. Furthermore, we prove that Radau‐IIA methods are stable for the subclass of equations where a = α + iγ with α, γ ∈ ℜ, γ sufficiently small, and b ∈ ∁.