Scalar Delay Differential Equation Research Articles

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.

Reduced ordered modelling of time delay systems using galerkin approximations and eigenvalue decomposition

In this paper, an r-dimensional reduced-order model (ROM) for infinite-dimensional delay differential equations (DDEs) is developed. The eigenvalues of the ROM match the r rightmost characteristic roots of the DDE with a user-specified tolerance of $$\varepsilon $$. Initially, the DDE is approximated by an N-dimensional set of ordinary differential equations using Galerkin approximations. However, only $$N_{c}$$$$(< N)$$ eigenvalues of this N-dimensional model match (with a tolerance of $$\varepsilon $$) the rightmost characteristic roots of the DDEs. By performing numerical simulations, an empirical relationship for $$N_{c}$$ is obtained as a function of N and $$\varepsilon $$ for a scalar DDE with multiple delays. Using eigenvalue decomposition, an r$$(= N_{c})$$ dimensional model is constructed. First, an appropriate r is chosen, and then the minimum value of N at which at least r roots converge is selected. For each of the test cases considered, the time and frequency responses of the original DDE obtained using direct numerical simulations are compared with the corresponding r- and N-dimensional systems. By judiciously selecting r, solutions of the ROM and DDE match closely. Next, an r-dimensional model is developed for an experimental 3D hovercraft in the presence of delay. The time responses of the r-dimensional model compared favorably with the experimental results.

Delay dependent stability of stochastic split-step θ methods for stochastic delay differential equations

In this paper, the delay dependent asymptotic mean square stability of the stochastic split-step θ method for a scalar linear stochastic delay differential equation with real coefficients is studied. The full stability region of this method is given by using root locus technique. The necessary and sufficient condition with respect to the equation coefficients, time stepsize and method parameter θ is derived. It is proved that the stochastic split-step backward Euler can preserve the asymptotic mean square stability of the underlying system completely. In addition, the numerical stability regions of the stochastic split-step θ method and the stochastic θ method are compared with each other. At last, we validate our conclusions by numerical experiments.

Positivity for non-Metzler systems and its applications to stability of time-varying delay systems

In 1950, Wazewski obtained the following necessary and sufficient condition for positivity of linear systems of ordinary differential equations: the matrix of coefficients is Metzler (i.e. off-diagonal elements in the matrix of coefficients are nonnegative). No results on the positivity of solutions to delay systems in the case where the matrix is non-Metzler were expected to be obtained. It was demonstrated that for delay systems the Wazewski condition is not a necessary one. We prove results on positivity of solutions for non-Metzler systems of delay differential equations. New explicit tests for exponential stability are obtained as applications of results on positivity for non-Metzler systems. Examples demonstrate possible applications of our theorems to stabilization. For instance, in view of our results, the implicit requirement on dominance of the main diagonal can be skipped. Our approach is based on nonoscillation of solutions and positivity of the Cauchy functions of scalar diagonal delay differential equations.

Hopf bifurcation analysis of scalar implicit neutral delay differential equation

Hopf bifurcation analysis of scalar implicit neutral delay differential equation

Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.

Stability results for nonlinear functional differential equations using fixed point methods

We present new conditions for stability of the zero solution for three distinct classes of scalar nonlinear delay differential equations. Our approach is based on fixed point methods and has the advantage that our conditions neither require boundedness of delays nor fixed sign conditions on the coefficient functions. Our work extends and improves a number of recent stability results for nonlinear functional differential equations in a unified framework. A number of examples are given to illustrate our main results.

Existence of Positive Periodic Solutions for Scalar Delay Differential Equations with and without Impulses

The paper is concerned with a broad family of scalar periodic delay differential equations with linear impulses, for which the existence of a positive periodic solution is established under very general conditions. The proofs rely on fixed point arguments, employing either the Schauder theorem or Krasnoselskii fixed point theorem in cones. The results are illustrated with applications to an impulsive hematopoiesis model or generalized Nicholson’s equations, among other selected examples from mathematical biology. The method presented here turns out to be very powerful, in the sense that the derived theorems largely generalize and improve other results in recent literature, even for the situation without impulses.

Oscillation of a linear delay differential equation with slowly varying coefficient

A new sufficient condition is given for the oscillation of all solutions of a linear scalar delay differential equation with slowly varying and uniformly positive coefficient at infinity. Based on the celebrated Myshkis type necessary condition, we show that this new sufficient condition is sharp within the considered class of delay differential equations.

Stability in a scalar differential equation with multiple, distributed time delays

We consider a linear scalar delay differential equation (DDE), consisting of two arbitrary distributed time delays. We formulate necessary conditions for stability of the trivial solution which are independent of the distributions. For the case of one discrete and one gamma distributed delay, we give an explicit description of the region of stability of the trivial solution and discuss how this depends on the model parameters.

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincar\'e section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds. This allows us to study transitions through resonance tongues and the breakup of a 1:4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.

On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability

This work is concerned with the study of the scalar delay differential equation \begin{document} $ z^{\prime\prime}(t)=h^2\,V(z(t-1)-z(t))+h\,z^\prime(t) $ \end{document} motivated by a simple car-following model on an unbounded straight line. Here, the positive real \begin{document} $h$ \end{document} denotes some parameter, and \begin{document} $V$ \end{document} is some so-called optimal velocity function of the traffic model involved. We analyze the existence and local stability properties of solutions \begin{document} $z(t)=c\,t+d$ \end{document} , \begin{document} $t∈\mathbb{R}$ \end{document} , with \begin{document} $c,d∈\mathbb{R}$ \end{document} . In the case \begin{document} $c\not=0$ \end{document} , such a solution of the differential equation forms a wavefront solution of the car-following model where all cars are uniformly spaced on the line and move with the same constant velocity. In particular, it is shown that all but one of these wavefront solutions are located on two branches parametrized by \begin{document} $h$ \end{document} . Furthermore, we prove that along the one branch all solutions are unstable due to the principle of linearized instability, whereas along the other branch some of the solutions may be stable. The last point is done by carrying out a center manifold reduction as the linearization does always have a zero eigenvalue. Finally, we provide some numerical examples demonstrating the obtained analytical results.

Stability Threshold for Scalar Linear Periodic Delay Differential Equations

AbstractWe prove that for the linear scalar delay differential equationwith non-negative periodic coefficients of period P &gt; 0, the stability threshold for the trivial solution is , assuming that b(t+1)−a(t) does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that, in general, r = 0 is not a stability threshold.

Configurations of periodic orbits for equations with delayed positive feedback

We consider scalar delay differential equations of the formx˙(t)=−μx(t)+f(x(t−1)), where μ>0 and f is a nondecreasing C1-function. If χ is a fixed point of fμ:R∋u↦f(u)/μ∈R with fμ′(χ)>1, then [−1,0]∋s↦χ∈R is an unstable equilibrium. A periodic solution is said to have large amplitude if it oscillates about at least two fixed points χ−<χ+ of fμ with fμ′(χ−)>1 and fμ′(χ+)>1. We investigate what type of large-amplitude periodic solutions may exist at the same time when the number of such fixed points (and hence the number of unstable equilibria) is an arbitrary integer N≥2. It is shown that the number of different configurations equals the number of ways in which N symbols can be parenthesized. The location of the Floquet multipliers of the corresponding periodic orbits is also discussed.

A positivity-based approach to delay-dependent stability of systems with large time-varying delays

In this paper we propose new explicit tests for positivity and exponential stability of systems with large time-varying delays. Our approach is based on nonoscillation of solutions of the corresponding diagonal scalar delay differential equations. Numerical examples illustrate the efficiency of the results.

On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model

We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the context of recent literature.

Stability and bifurcation analysis of a generalized scalar delay differential equation.

This paper deals with the stability and bifurcation analysis of a general form of equation D(α)x(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.

Tracking the Algebraic Multiplicity of Crossing Imaginary Roots for Generic Quasipolynomials: A Vandermonde-Based Approach

A standard approach in analyzing dynamical systems consists in identifying and understanding the eigenvalues bifurcations when crossing the imaginary axis. Efficient methods for crossing imaginary roots identification exist. However, to the best of the author's knowledge, the multiplicity of such roots was not deeply investigated. In recent papers by the authors [1], [2], it is emphasized that the multiplicity of the zero spectral value can exceed the number of the coupled scalar delay-differential equations and a constructive approach Vandermonde-based allowing to an adaptive bound for such a multiplicity is provided. Namely, it is shown that the zero spectral value multiplicity depends on the system structure (number of delays and number of non zero coefficients of the associated quasipolynomial) rather than the degree of the associated quasipolynomial [3]. This technical note extends the constructive approach in investigating the multiplicity of crossing imaginary roots $j\omega$ where $\omega\neq 0$ and establishes a link with a new class of functional confluent Vandermonde matrices. A symbolic algorithm for computing the LU-factorization for such matrices is provided. As a byproduct of the proposed approach, a bound sharper than the Polya-Szego generic bound arising from the principle argument is established.