General Delay Differential Equations Research Articles

Some New Oscillation Criteria of Even-Order Quasi-Linear Delay Differential Equations with Neutral Term

The neutral delay differential equations have many applications in the natural sciences, technology, and population dynamics. In this paper, we establish several new oscillation criteria for a kind of even-order quasi-linear neutral delay differential equations. Comparing our results with those in the literature, our criteria solve more general delay differential equations with neutral type, and our results expand the range of neutral term coefficient. Some examples are given to illustrate our conclusions.

Hyers–Ulam stability of linear functional differential equations

In this paper, the stability of some classes of linear functional differential equations was discussed by direct method, iteration method, fixed point method and open mapping theorem. It is shown that the Hyers–Ulam stability holds true for y(n)=g(t)y(t−τ)+h(t). The stability of functional differential equations with multiple delays of first order and general delay differential equations also have been discussed.

Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations.

In pharmacokinetics/pharmacodynamics (PKPD) the measured response is often delayed relative to drug administration, individuals in a population have a certain lifespan until they maturate or the change of biomarkers does not immediately affects the primary endpoint. The classical approach in PKPD is to apply transit compartment models (TCM) based on ordinary differential equations to handle such delays. However, an alternative approach to deal with delays are delay differential equations (DDE). DDEs feature additional flexibility and properties, realize more complex dynamics and can complementary be used together with TCMs. We introduce several delay based PKPD models and investigate mathematical properties of general DDE based models, which serve as subunits in order to build larger PKPD models. Finally, we review current PKPD software with respect to the implementation of DDEs for PKPD analysis.

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.

Equivalence of the MTS Method and CMR Method for Differential Equations Associated with Semisimple Singularity

In this paper, the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method is proved for computing the normal forms of ordinary differential equations and delay differential equations. The delay equations considered include general delay differential equations (DDE), neutral functional differential equations (NFDE) (or neutral delay differential equations (NDDE)), and partial functional differential equations (PFDE). The delays involved in these equations can be discrete or distributed. Particular attention is focused on dynamics associated with the semisimple singularity, and both the MTS and CMR methods are applied to compute the normal forms near the semisimple singular point. For the ordinary differential equations (ODE), we show that the two methods are equivalent up to any order in computing the normal forms; while for the differential equations with delays, we obtain the conditions under which the normal forms, derived by using the MTS and CMR methods, are identical up to third order. Different types of practical examples with delays are presented to demonstrate the application of the theoretical results, associated with Hopf, Hopf-zero and double-Hopf singularities.

STEADY-STATE, HOPF AND STEADY-STATE-HOPF BIFURCATIONS IN DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO A DAMPED HARMONIC OSCILLATOR WITH DELAY FEEDBACK

In this paper, employing the normal form theory of delay differential equations due to Faria and Magalhães, we present explicit formulas of the coefficients of a normal form associated with the flow on a center manifold with the unfolding for general delay differential equations under the cases of steady-state, Hopf and steady-state-Hopf singularities. The explicit conditions determining the transcritical and pitchfork bifurcations for steady-state singularity, determining the direction and stability of Hopf bifurcations, and determining the coefficients of a normal form with universal unfolding for steady-state-Hopf singularity up to third order are obtained. Using the obtained results, we give a complete description of bifurcation scenario of the damped harmonic oscillator with delay feedback near the zero equilibrium. Finally, numerical simulations are given to illustrate our theoretical results and some numerical extensions are obtained as a supplement to our theoretical analysis.

Generalized Hopf bifurcation in delay differential equations

Here we employ the Lyapunov-Schmidt procedure to investigate bifurcations in a general delay differential equation when the infinitesimal generator has, for a critical value of the parameter, a pair of nonsemisimple purely imaginary eigenvalues with multiplicity k . We derive criteria, explicitly in terms of the systems parameter values, for the existence of bifurcating periodic solutions and for the description of the bifurcation direction. The general result is illustrated by a detailed case study of the van del Pole oscillator.

An efficient unified approach for the numerical solution of delay differential equations

In this paper we propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a standard step-by-step approach, such as Runge-Kutta or linear multi-step methods, and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. Using this interpretation we develop an efficient technique to solve the resulting discontinuous IVP. We also give a more clear process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. The new modular design for the resulting simulator we introduce, helps to accelerate the utilization of advances in the different components of an effective numerical method. Such components include the underlying discrete formula, the interpolant for dense output, the strategy for handling discontinuities and the iteration scheme for solving any implicit equations that arise.

On global properties of time-delayed feedback control: weakly nonlinear analysis

We address the problem which type of initial state yields successful stabilisation when time-delayed feedback control is applied. Using well known concepts from bifurcation theory we provide a general mechanism that limits the corresponding basins of attraction. Application of normal form analysis to general delay-differential equations proves the universality of our results. The empirical criteria which determine the prospective size of the basin are easily accessible and a demonstration in terms of a simple numerical simulation is provided.

The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag

This paper is concerned with the exponential stability of singularly perturbed delay differential equations with a bounded (state-independent) lag. A generalized Halanay inequality is derived in Section 2, and in Section 3 a sufficient condition will be provided to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is exponentially stable uniformly for sufficiently small ε>0. This type of exponential asymptotic stability can obviously be applied to general delay differential equations with a bounded lag.

Contractivity of continuous Runge-Kutta methods for delay differential equations

In this paper the author investigates the stability of numerical methods for general delay differential equations of the type ▪ where α( t) ≤ t and y( t) is a vector complex-valued function. Contractivity conditions are found for Runge-Kutta methods as applied to linear and nonlinear scalar equations. As for systems, a general condition is found for the contractivity of the solution of (1) in any vector norm, and a numerical method is proposed which preserves the contractivity in the maximum norm.

Quantum theory of an atom near partially reflecting walls.

We consider first a dielectric medium of identical two-state atoms coupled by the radiation field to an initially excited atom outside the dielectric. From the Schr\"odinger equation follows a delay-differential equation describing how the atom interacts with the dielectric by virtual photon exchanges. In the macroscopic limit of a continuous distribution of atoms in the dielectric, we derive a simpler delay-differential equation in which a Fresnel reflection coefficient appears. We apply our results to a model of an atom in a multimode Fabry-Perot resonator, and obtain a general delay-differential equation for the probability amplitude of the initially excited state. This equation predicts well-known Rabi oscillations when the round-trip photon propagation time is negligible compared with the inverse of the Rabi frequency and the mirrors are highly reflective. For low mirror reflectivities we recover Purcell's prediction that the emission rate is enhanced by the cavity Q factor. When the photon bounce time is large compared with the inverse Rabi frequency, Rabi oscillations do not occur. We discuss the Ewald-Oseen extinction theorem from the standpoint of quantum mechanics.