Narrow Results

In this paper, we provide new necessary and sufficient conditions of the asymptotic stability for a class of quasilinear differential equations with several delays and oscillating coefficients. Our results are established by means of fixed point theory and improve those obtained in [J. R. Graef, C. Qian and B. Zhang, Asymptotic behavior of solutions of differential equations with variable delays, Proc. London Math. Soc.81 (2000) 72–92; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal.63 (2005) e233–e242].

Traditional epidemic models consider that individual processes occur at constant rates; an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption indeed fails for almost all infectious diseases, in which the infection time usually follows a probability distribution more or less spread around a mean value. We presented some years ago a general treatment for an SIRS model in which both the infected and the immune phases admit such a description, where the system shows transitions between endemic and oscillating situations. In the present contribution, we consider the effect of seasonality on the transmission rate. Performing numerical simulations of the delayed equations plus seasonality, we have found a rich and complex scenario of oscillations that could be relevant in real epidemics.

Stabilization of a class of linear time-delay systems can be achieved by a numerical procedure, called the continuous pole placement method [Michiels et al., 2000]. This method can be seen as an extension of the classical pole placement algorithm for ordinary differential equations to a class of delay differential equations. In [Michiels et al., 2000] it was applied to the stabilization of a linear time-invariant system with an input delay using static state feedback. In this paper we study the limitations of such delayed state feedback laws. More precisely we completely characterize the class of stabilizable plants in the 2D-case. For that purpose we make use of numerical continuation techniques. The use of delayed state feedback in various control applications and the effect of its limitations are briefly discussed.

This is a survey on discretizing delay equations from a geometric-qualitative view-point. Concepts like compact attractors, hyperbolic periodic orbits, the saddle structure around hyperbolic equilibria, center-unstable manifolds of equilibria, inertial manifolds, structural stability, and Kamke monotonicity are considered. Error estimates for smooth and nonsmooth initial data in various C^j topologies are provided. The emphasis is put on Runge–Kutta methods with natural interpolants. The paper ends with a collection of the related results on retarded functional differential equations with bounded delay.

A ring of discrete-time delayed neural network with self-feedback and a valid nonmonotonic activation function is explored. Coexistence of multiple stable equilibria and chaotic dynamics are demonstrated in this discrete dynamical system. Specifically, 2^m stable stationary solutions and their basins of attraction are found for a loop with m-neurons network. The theory is established by formulating parameter conditions according to a geometric observation. The networks are further confirmed to exhibit chaotic dynamics when the magnitudes of inhibitory self-feedback weights are large enough. The scenario is based on building a snapback repeller and Marotto's theorem.

The paper studies the modeling of time series with the prescribed dependence of the volatility on the sampling frequency. This dependence is often observed for financial time series. We suggest to model the dependence of volatility on sampling frequency via delay equations for the underlying prices. It appears that these equations allow to model the price processes with volatility that increases when the sampling rates increase. In addition, these equations are able to model the inverse phenomena where the volatility decreases with the increase in sampling frequencies.

A linear difference-differential system with constant coefficients and containing a small parameter is investigated. We establish sufficient conditions for the existence of an asymptotic integral manifold of solutions described by a linear system of differential equations without delays.

This is a survey on discretizing delay equations from a geometric-qualitative view-point. Concepts like compact attractors, hyperbolic periodic orbits, the saddle structure around hyperbolic equilibria, center-unstable manifolds of equilibria, inertial manifolds, structural stability, and Kamke monotonicity are considered. Error estimates for smooth and nonsmooth initial data in various C^j topologies are provided. The emphasis is put on Runge–Kutta methods with natural interpolants. The paper ends with a collection of the related results on retarded functional differential equations with bounded delay.

A method of formalizing the analysis of asymptotic properties of solutions to systems of differential equations with distributed time-delays and Boolean-type nonlinearities is offered. Such objects arise in many applications, but of most importance are systems coming from gene regulatory networks (GRN). The dynamics of GRN are governed by sigmoid-type nonlinearities which are close to the step functions. This is due to the fact that genes are only activated if certain concentrations are close to the respective threshold values. The delay effects arise from the time required to complete transcription, translation and diffusion to the place of action of a protein.

We describe an algorithm of localizing stationary points in the presence of delays as well as stability analysis around such points. The basic technical tool consists in replacing step functions with the so-called "logoid functions", combined with a special modification of the well-known "linear chain trick", and investigating the smooth systems thus obtained.

A significant part of this framework is based on asymptotic analysis of singularly perturbed matrices, where we apply Mathematica to be able to derive exact stability criteria. This work is a brief review of the results presented in Ref. 1.