Scalar Delay Differential Equation Research Articles

On the structure of the infinitesimal generators of scalar one-dimensional semigroups with discrete Lyapunov functionals

AbstractDynamical systems generated by scalar reaction-diffusion equations on an interval enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system. Other systems, like the linear scalar delay differential equations under monotone feedback conditions, possess similar kinds of discrete Lyapunov functions. Here we discuss and characterize classes of linear equations that generate semiflows acting on $$C^0[0,1]$$ C 0 [ 0 , 1 ] or on $$C^1[0,1]$$ C 1 [ 0 , 1 ] which admit discrete Lyapunov functions related to the zero number. We show that, if the space is $$C^1[0,1]$$ C 1 [ 0 , 1 ] , the corresponding equations are essentially parabolic partial differential equations. In contrast, if the space is $$C^0[0,1]$$ C 0 [ 0 , 1 ] , the corresponding equations are generalizations of monotone feedback delay differential equations.

An improved result on bounded/periodic solutions for some scalar delay differential equations

An improved result on bounded/periodic solutions for some scalar delay differential equations

Bounded/periodic solutions for a class of scalar delay differential equations

In this paper we use upper and lower solutions, the monotone iterative method, and the Schauder fixed point theorem to establish a new result on the existence of bounded/periodic solutions over R \mathbb R for a class of scalar delay differential equations of the form d u d t = f ( t , u ( t ) , u ( t − r 1 ) , ⋯ , u ( t − r m ) ) \begin{equation*} \frac {du}{dt}=f(t,u(t), u(t-r_1), \cdots , u(t-r_m)) \end{equation*} with constant delays, where f f satisfies a one-sided Lipschitz condition on the state variables u ( t ) u(t) , u ( t − r 1 ) u(t-r_1) , …, u ( t − r m ) u(t-r_m) . Applications of this result to some population models are given.

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

Global exponential stability analysis for tick population model with delayed sub-linear Gamma-Ricker nonlinearities

We argue that self-regulation in tick population dynamics in the natural world is not associated with the egg-production saturation, that was normally assumed in existing literature and modeled using the Ricker reproduction function. We argue that the nonlinearity and self-regulatory inhibition appear in the synergistic effect of tick cooperation and the crowding within groups, and an appropriate nonlinearity can be the sub-linear Gamma-Ricker function. We also noticed that the diapaused development leads to multiple time lags in a structured tick population dynamics model, leading to a scalar delay differential equation with two time lags appearing in the sub-linear Gamma-Ricker nonlinearity. This model has always one and only one positive equilibrium. Here we develop sharp conditions for this model to exhibit persistence and global convergence behaviors.

Non-autonomous Differential Systems with Delays: A Global Attraction Analysis

In this paper, we derive criteria of global attractivity of a (possibly constant) positive periodic solution in non-autonomous systems of delay differential equations. Our approach can be viewed as the extension for non-autonomous systems of the folkloric connection between discrete dynamics and scalar delay differential equations. It is worth mentioning that we provide delay-dependent criteria of global attraction that cover the best delay independent conditions. We apply our results to non-autonomous variants of several classical models such that Nicholson’s blowfly equation, Goodwin’s model oscillator, the Mackey–Glass equation and systems with patch structure.

On oscillation of solutions of scalar delay differential equation in critical case

In this paper we study the oscillation problem for the known scalar delay differential equation. We assume that the coefficients of this equation have an oscillatory behaviour with an amplitude of oscillation tending to zero at infinity. The asymptotic formulae for the solutions of the considered equation in the so-called critical case are constructed. We give the conditions for existence of oscillatory or nonoscillatory solutions in terms of certain numerical quantities. The obtained results are illustrated by a number of examples.

CONSTRUCTION OF STABILITY DOMAINS FOR LINEAR DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS

The aim of the present article is to investigate of solutions stability of linear autonomous differential equations with retarded argument. The investigation of stability can be reduced to the root location problem for the characteristic equation. For the linear differential equation with several delays it is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation equation to have negative real part (and hence the zero solution to be asymptotically stable). For the scalar delay differential equation $$ \frac{dz}{dt}=c z(t) + a_1 z(t-1) + a_2 z(t-2) + ... + a_n z(t-n), $$ with fixed $c$, $c \in \mathbb{R}$, $a_k \in \mathbb{R}$, $1 \leq k \leq n$, stability domains in the parameter plane are obtained. We investigate the boundedness conditions and construct a domain of stability for linear autonomous differential equation with several delays. We use D-partition method, argument principle and numerical methods to construct of stability domains.

Insights into the multiplicity-induced-dominancy for scalar delay-differential equations with two delays

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems of retarded or neutral type with a single delay, if a root of its characteristic equation attains its maximal multiplicity, then this root is the rightmost spectral value, and hence it determines the exponential behavior of the system, a property usually referred to as multiplicity-induced-dominancy (MID). In this paper, we investigate the MID property for one of the simplest cases of systems with two delays, a scalar delay-differential equation of first order with two delayed terms of order zero. We discuss the standard approach based on the argument principle for establishing the MID property for single-delay systems and some of its limitations in the case of our simple system with two delays, before proposing a technique based on crossing imaginary roots that allows to conclude that the MID property holds in our setting.

Distributed Delay Differential Equation Representations of Cyclic Differential Equations

Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the well-known linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with non-linear transit rates and possibly delayed arguments are also equivalent to a scalar distributed delay differential equation. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute the characteristic function-- without calculating a determinant. We derive the equivalent scalar DDE for two examples of models in mathematical biology and use the DDE formulation to identify physiological processes that were otherwise hidden by the compartmental structure of the ODE model.

Single species population dynamics in seasonal environment with short reproduction period

We present a periodic nonlinear scalar delay differential equation model for a population with short reproduction period. By transforming the equation to a discrete dynamical system, we reduce the infinite dimensional problem to one dimension. We determine the basic reproduction number not merely as the spectral radius of an operator, but as an explicit formula and show that is serves as a threshold parameter for the stability of the trivial equilibrium and for permanence.

Complete dimensional collapse in the continuum limit of a delayed SEIQR network model with separable distributed infectivity.

We take up a recently proposed compartmental SEIQR model with delays, ignore loss of immunity in the context of a fast pandemic, extend the model to a network structured on infectivity and consider the continuum limit of the same with a simple separable interaction model for the infectivities beta . Numerical simulations show that the evolving dynamics of the network is effectively captured by a single scalar function of time, regardless of the distribution of beta in the population. The continuum limit of the network model allows a simple derivation of the simpler model, which is a single scalar delay differential equation (DDE), wherein the variation in beta appears through an integral closely related to the moment generating function of u=sqrt{beta }. If the first few moments of u exist, the governing DDE can be expanded in a series that shows a direct correspondence with the original compartmental DDE with a single beta . Even otherwise, the new scalar DDE can be solved using either numerical integration over u at each time step, or with the analytical integral if available in some useful form. Our work provides a new academic example of complete dimensional collapse, ties up an underlying continuum model for a pandemic with a simpler-seeming compartmental model and will hopefully lead to new analysis of continuum models for epidemics.

Multi-cycle Periodic Solutions of a Differential Equation with Delay that Switches Periodically

We describe the behaviour of solutions of a scalar Delay Differential Equation (DDE) with delay that periodically switches between two constant values. Such an equation arises naturally from structured vector populations involved in a range of vector-borne diseases spreading in a periodically varying environment. We examine if and how the two different time lags and the switching time influence the existence and patterns of periodic solutions. We pay particular attention to the patterns involving multi-cycles within the prime period of the periodic solutions.

Global Continuation of Periodic Oscillations to a Diapause Rhythm

We consider a scalar delay differential equation $$\dot{x}(t)=-dx(t)+f((1-\alpha )\rho x(t-\tau )+\alpha \rho x(t-2\tau ))$$ with an instant mortality rate $$d>0$$, the nonlinear Rick reproductive function f, a survival rate during all development stages $$\rho $$, and a proportion constant $$\alpha \in [0, 1]$$ with which population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within $$(3\tau , 6\tau )$$ for a wide range of parameter values. We show this existence of periodic solutions not only for the delay $$\tau $$ near the first critical value $$\tau ^*$$ when a local Hopf bifurcation takes place near the positive equilibrium, but for all $$\tau >\tau ^*$$. We obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li–Muldowney technique to rule out periodic solutions with period $$3\tau $$. We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium.

StabilIty of Scalar Nonlinear Fractional Differential Equations with Linearly Dominated Delay

Abstract In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved under a small nonlinear Lipschitz perturbation of the fractional delay differential equation.

Global asymptotic stability for a periodic delay hematopoiesis model with impulses

In this paper, sufficient conditions for the global asymptotic stability of a broad family of periodic impulsive scalar delay differential equations are obtained. These conditions are applied to a periodic hematopoiesis model with multiple time-dependent delays and linear impulses, in order to establish criteria for the global asymptotic stability of a positive periodic solution. The present results are discussed within the context of recent literature. In conclusion, when compared with previous works, not only sharper stability criteria are obtained here, even for models without impulses, but also the usual constraints imposed on the linear impulses are relaxed.

Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems

In recent works, exact and nonstandard finite difference schemes for scalar first order linear delay differential equations have been proposed. The aim of the present work is to extend these previous results to systems of coupled delay differential equations X ′ ( t ) = A X ( t ) + B X ( t - τ ) , where X is a vector, and A and B are commuting real matrices, in general not simultaneously diagonalizable. Based on a constructive expression for the exact solution of the vector equation, an exact scheme is obtained, and different nonstandard numerical schemes of increasing order are proposed. Dynamic consistency properties of the new nonstandard schemes are illustrated with numerical examples, and proved for a class of methods.

Laminar Chaos in Experiments: Nonlinear Systems with Time-Varying Delays and Noise.

A new type of dynamics called laminar chaos was recently discovered through a theoretical analysis of a scalar delay differential equation with time-varying delay. Laminar chaos is a low-dimensional dynamics characterized by laminar phases of nearly constant intensity with periodic durations and a chaotic variation of the intensity from one laminar phase to the next laminar phase. This is in stark contrast to the typically observed higher-dimensional turbulent chaos, which is characterized by strong fluctuations. In this Letter we provide the first experimental observation of laminar chaos by studying an optoelectronic feedback loop with time-varying delay. The noise inherent in the experiment requires the development of a nonlinear Langevin equation with variable delay. The results show that laminar chaos can be observed in higher-order systems, and that the phenomenon is robust to noise and a digital implementation of the variable time delay.

The first passage problem for stable linear delay equations perturbed by power law Lévy noise.

This article studies a linear scalar delay differential equation subject to small multiplicative power tail Lévy noise. We solve the first passage (the Kramers) problem with probabilistic methods and discover an asymptotic loss of memory in this non-Markovian system. Furthermore, the mean exit time increases with the power of the small noise amplitude, whereas the prefactor accounts for memory effects. In particular, we discover a nonlinear delay-induced exit acceleration due to a non-normal growth phenomenon. Our results are illustrated by the example of a linear delay oscillator driven by α-stable Lévy flights.

Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.