Scalar Delay Equations Research Articles

On the compactness of solutions to certain operator inequalities arising from the Likhtarnikov — Yakubovich frequency theorem

We study the compactness property of operator solutions to certain operator inequalities arising from the frequency theorem of Likhtarnikov — Yakubovich for C0-semigroups. We show that the operator solution can be described through solutions of an adjoint problem as it was previously known under some regularity condition. Thus we connect some regularity properties of the semigroup with the compactness of the operator in the general case. We also prove several results useful for checking the non-compactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases arising in applications. As an example, we apply these theorems for a scalar delay equation posed in a proper Hilbert space and show that the operator solution cannot be compact. This results are related to the author recent work on a non-local reduction principle of cocycles (non-autonomous dynamical systems) in Hilbert spaces.

Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay

We study scalar delay equations \begin{document}$ \dot{x} (t) = \lambda f(x(t-1)) + b^{-1} (x(t) + x(t -p/2)) $\end{document} with odd nonlinearity $ f $, real nonzero parameters $ \lambda, \, b $, and two positive time delays $ 1, \ p/2 $. We assume supercritical Hopf bifurcation from $ x \equiv 0 $ in the well-understood single-delay case $ b = \infty $. Normalizing $ f' (0) = 1 $, branches of constant minimal period $ p_k = 2\pi/\omega_k $ are known to bifurcate from eigenvalues $ i\omega_k = i(k+\tfrac{1}{2})\pi $ at $ \lambda_k = (-1)^{k+1}\omega_k $, for any nonnegative integer $ k $. The unstable dimension of these rapidly oscillating periodic solutions is $ k $, at the local branch $ k $. We obtain stabilization of such branches, for arbitrarily large unstable dimension $ k $, and for, necessarily, delicately narrow regions $ \mathcal{P} $ of scalar control amplitudes $ b < 0 $.For $ p $: = $ p_k $ the branch $ k $ of constant period $ p_k $ persists as a solution, for any $ b\neq 0 $. Indeed the delayed feedback term controlled by $ b $ vanishes on branch $ k $: the feedback control is noninvasive there. Following an idea of Pyragas [30], we seek parameter regions $ \mathcal{P} = (\underline{b}_k, \overline{b}_k) $ of controls $ b \neq 0 $ such that the branch $ k $ becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for $ \mathcal{P} $ in the limit of large $ k $. Our analysis is based on a 2-scale covering lift for the slow and rapid frequencies involved.These results complement earlier results in [8] which required control terms \begin{document}$ b^{-1} (x(t-\vartheta) + x(t-\vartheta -p/2)) $\end{document} with a third delay $ \vartheta $ near 1.

Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system

A question of ongoing interest for linear Time-delay systems is to determine conditions on the equation parameters that guarantee the exponential stability of solutions. In recent works a new interesting property of time-delay systems was emphasized. As a matter of fact, the multiple spectral values for time-delay systems was characterized by using a Birkhoff/Vandermonde-based approach. Then, a multiplicity induced stability criteria were exhibited for reduced order systems; scalar delay-equations and a special class of second order systems. This work, further explores such a criteria and shows their applicability to the control of a mechanical system.

Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

We embark on a case study for the scalar delay equation $$\begin{aligned} \dot{x} (t) = \lambda f(x(t-1)) + b^{-1} (x(t-\vartheta ) + x(t-\vartheta -p/2)) \end{aligned}$$ with odd nonlinearity f, real nonzero parameters \(\lambda , \, b\), and three positive time delays \(1,\, \vartheta ,\, p/2\). We assume supercritical Hopf bifurcation from \(x \equiv 0\) in the well-understood single-delay case \(b = \infty \). Normalizing \(f' (0)=1\), branches of constant minimal period \(p_k = 2\pi /\omega _k\) are known to bifurcate from eigenvalues \(i\omega _k = i(k+\tfrac{1}{2})\pi \) at \(\lambda _k = (-1)^{k+1}\omega _k\), for any nonnegative integer k. The unstable dimension is k, at the local branch k. We obtain stabilization of such branches, for arbitrarily large unstable dimension k. For \(p:= p_k\) the branch k of constant period \(p_k\) persists as a solution, for any \(b\ne 0\) and \(\vartheta \ge 0\). Indeed the delayed feedback term controlled by b vanishes on branch k: the feedback control is noninvasive there. Following an idea of Pyragas, we seek parameter regions \(\mathcal {P}\) of controls \(b \ne 0\) and delays \(\vartheta \ge 0\) such that the branch k becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for \(\mathcal {P}\) in the limit of large k. The only two regions \(\mathcal {P} = \mathcal {P}^\pm \) which we were able to detect, in this setting, required delays \(\vartheta \) near 1, controls b near \((-1)^k \cdot 2/\omega _k\), and were of very small area of order \(k^{-4}\). Our analysis is based on a 2-scale covering lift for the frequencies involved.

The Mackey–Glass model of respiratory dynamics: Review and new results

Since the celebrated Mackey–Glass model of respiratory dynamics was introduced in 1977, many results on its qualitative behavior have been obtained, including oscillation, stability and chaos. The paper reviews some known properties and presents new results for more general models: equations with time-dependent parameters, several delays, a positive periodic equilibrium and distributed delays. The problems considered in the paper involve existence, positivity and permanence of solutions, oscillation and global asymptotic stability. In addition, some general approaches to the study of nonlinear nonautonomous scalar delay equations are outlined. The paper generalizes and unifies existing results and provides an outlook on further studies.

Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedback

In many situations in physics, engineering and biology time delays arise naturally due to the time needed to transport information from one part of the system to another and/or to react to incoming information. When differential equations are used in the mathematical modelling, then incorporating time delays leads to a description by a delay differential equation. We consider here a class of second-order scalar delay equations without instantaneous feedback, where the delays enter according to a distribution function. This is a natural description whenever there is more than one delay. In this article we show that for this class of systems one can derive stability information about the distributed-delay system by considering the single-delay system where the delay is the mean delay of the distribution function. More specifically, we prove that the asymptotic stability of the zero solution of the second-order delay equation with symmetric delay distribution is implied by the stability of the associated mean-delay equation. Our proof is based on the comparison of stability charts of the two equations.

Stability in discrete equations with variable delays

In this paper we study the stability of the zero solution of difference equations with variable delays. In particular we consider the scalar delay equation

Stabilizing effect of delay distribution on second-order scalar delay equations

In this paper we investigate the effect of delay distribution on a type of second-order scalar delay differential equation. More specifically, we compare the stability region of the equations with one and two delays. Our findings indicate an increase in the stability region when the delay distribution satisfies certain conditions. We use these ideas to argue for considering the computational time delay as a stabilizing factor in hybrid testing.

On the global dynamics of attractors for scalar delay equations

A semi-conjugacy from the dynamics of the global attractors for a family of scalar delay differential equations with negative feedback onto the dynamics of a simple system of ordinary differential equations is constructed. The construction and proof are done in an abstract setting, and hence, are valid for a variety of dynamical systems which need not arise from delay equations. The proofs are based on the Conley index theory.

Stability preserving spline approximations for scalar functional differential equations

We consider a method for constructing uniformly stable spline approximations for scalar delay equations.

The global existence of solutions of delay differential equations

Recent studies of differential equations with delayed arguments have revealed interesting, and often surprising, facts about the behavior of solutions. In this paper, the question of the existence of solutions defined for all time is investigated. We begin by applying the following result of Winston [5] to demonstrate some simple scalar delay equations which admit only the zero solution on the line.