Dichotomy results for delay differential equations with negative Schwarzian

We gain further insight into the use of the Schwarzian derivative to obtain new results for a family of functional differential equations including the famous Wright's equation and the Mackey-Glass type delay differential equations. We present some dichotomy results, which allow us to get easily computable bounds of the global attractor. We also discuss related conjectures, and formulate new open problems.Comment: 16 pages, submitted to Chaos,Solitons,Fractal

Asymptotic methods for delay equations.

Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006

Controlling Mackey--Glass chaos

The Mackey--Glass equation, which was proposed to illustrate nonlinear phenomena in physiological control systems, is a classical example of a simple looking time delay system with very complicated behavior. Here we use a novel approach for chaos control: we prove that with well chosen control parameters, all solutions of the system can be forced into a domain where the feedback is monotone, and by the powerful theory of delay differential equations with monotone feedback we can guarantee that the system is not chaotic any more. We show that this domain decomposition method is applicable with the most common control terms. Furthermore, we propose an other chaos control scheme based on state dependent delays.Comment: accepted in Chaos: An Interdisciplinary Journal of Nonlinear Scienc

Mackey-Glass type delay differential equations near the boundary of absolute stability

For equations $x'(t) = -x(t) + \zeta f(x(t-h)), x \in \R, f'(0)= -1, \zeta > 0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to zero when both $\zeta -1 >0$ and $h(\zeta-1)^{1/8}$ are less than some constant (independent on $h,\zeta$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.Comment: 16 pages, 1 figure, accepted for publication in the Journal of Mathematical Analysis and Application

The role of the central chemoreceptor in causing periodic breathing.

In a previous publication (Fowler et aL, 1993), we reduced the classical cardiorespiratory control model of (Grodins et aL, 1967) to a much simpler form, which we then used to study the phenomenon of periodic breathing. In particular, cardiac output was assumed constant, and a single (constant) delay representing arterial blood transport time between lung and brain was included in the model. In this paper we extend this earlier work, both by allowing for the variability in transport delays, due to the dependence of cardiac output on blood gas concentrations, and also by including further delays in the system. In addition, we extensively discuss the physiological implications of parameter variations in the model; several novel mechanisms for periodic breathing in clinical situations are proposed. The results are discussed in the light of recent observational studies. Keywords: Periodic breathing; Cheyne-Stokes respiration; heart-rate variability*, differential-delay equations. 1

Shape of transition layers in a differential--delay equation

We use asymptotic techniques to describe the bifurcation from steady-state to a periodic solution in the singularly perturbed delayed logistic equation ?x?(t) = ?x(t)+ ? f(x(t ? 1)) with ? ? 1. The solution has the form of plateaus of approximately unit width separated by narrow transition layers. The calculation of the period two solution is complicated by the presence of delay terms in the equation for the transition layers, which induces a phase shift that has to be calculated as part of the solution. High order asymptotic calculations enable both the shift and the shape of the layers to be determined analytically, and hence the period of the solution. We show numerically that the form of transition layers in the four-cycles is similar to that of the two-cycle, but that a three-cycle exhibits different behaviour

On the Global Attractor of Delay Differential Equations with Unimodal Feedback

We give bounds for the global attractor of the delay differential equation $x'(t) =-\mu x(t)+f(x(t-\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the d5A5Aelay is sufficiently small, then all solution enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.Comment: 10 pages, submitted to Discrete and Continuous Dynamical Systems-Series A (DCDS

An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays

AbstractThis paper deals with singularly perturbed boundary value problem for a linear second order delay differential equation. It is known that the classical numerical methods are not satisfactory when applied to solve singularly perturbed problems in delay differential equations. In this paper we present an exponentially fitted finite difference scheme to overcome the drawbacks of the corresponding classical counter parts. The stability of the scheme is investigated. The proposed scheme is analyzed for convergence. Several linear singularly perturbed delay differential equations have been solved and the numerical results are presented to support the theory