Sjoerd Verduyn Lunel

The mathematics of delay equations with an application to the Lang-Kobayashi equations

For dynamical systems governed by feedback laws, time delays arise naturally in the feedback loop to represent effects due to communication, transmission, transportation or iternia effects. The introduction of time delays in a system of differential equations results in an infinite dimensional state space. We give an introduction to the basic theory of delay equations. After this introduction we illustrate the theory with a local analysis of the Lang-Kobayashi equations describing a laser with delayed optical feedback.

Spectral theory for delay equations

For dynamical systems governed by feedback laws, time delays arise naturally in the feedback loop to represent effects due to communication, transmission, transportation or inertia effects. The introduction of time delays in a system of differential equations results in an infinite dimensional state space. The solution operator associated with a differential delay equation is a nonself-adjoint operator defined on a Banach space. This implies that general abstract theorems cannot directly be applied. In this paper we discuss the spectral properties of autonomous and periodic differential delay equations, series expansions of solutions, completeness of eigenvectors and generalized eigenvectors, and solutions of delay equations that decay faster than any exponential.

Fixed points and pth moment exponential stability of stochastic delayed recurrent neural networks with impulses

Abstract New sufficient conditions for p th moment exponential stability of a class of impulsive stochastic delayed recurrent neural networks are presented by using fixed point theory. Our results neither require the boundedness, monotonicity and differentiability of the activation functions nor differentiability of the time varying delays. A class of impulsive delayed neural networks without stochastic perturbations are also considered. An example is given to illustrate our main results.

Characterising small solutions in delay differential equations through numerical approximations

The existence of small solutions, that is solutions that decay faster than any exponential, for linear time-dependent delay differential equations with bounded coefficients depends on specific properties of the coefficients. Although small solutions do not occur in the finite dimensional approximations of the delay differential equation we show that the existence of small solutions for delay differential equations can be predicted from the behaviour of the spectrum of the finite dimensional approximations.

Positive operators and Hausdorff dimension of invariant sets

In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are “infinitesimal similitudes” on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r(Lσ) = 1, where Lσ, σ > 0, is a naturally associated family of positive linear operators and r(Lσ) denotes the spectral radius of Lσ. We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

Spectral Theory for Neutral Delay Equations with Applications to Control and Stabilization

For dynamical systems governed by feedback laws, time delays arise naturally in the feedback loop to represent effects due to communication, transmission, transportation or iternia effects. The introduction of time delays in a system of differential equations results in an infinite dimensional state space. The solution operator associated with a differential delay equation is a nonself-adjoint operator defined on a Banach space. This implies that general abstract theorems cannot directly be applied. In this paper we discuss the spectral properties of neutral differential delay equations, series expansions of solutions, completeness of eigenvectors and generalized eigenvectors, solutions of neutral delay equations that decay faster than any exponential, and applications to control and stabilization.

Asymptotic estimates for the periods of periodic points of non-expansive maps

For each positive integer n we use the concept of ‘admissible arrays on n symbols’ to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In earlier joint work with M. Scheutzow, it was shown that the set Q(n) is intimately connected to the set of periods of periodic points of classes of nonexpansive nonlinear maps defined on the positive cone in R. In this paper we continue the characterization of Q(n) and present precise asymptotic estimates for the largest element of Q(n). For example, if γ (n) denotes the largest element of Q(n), then we show that limn→∞(n log n)−1/2 log γ (n) = 1. We also discuss why understanding further details about the fine structure of Q(n) involves some delicate number theoretical issues.

Control by time delayed feedback near a hopf bifurcation point

In this paper we study the stabilization of rotating waves using time delayed feedback control. It is our aim to put some recent results in a broader context by discussing two different methods to determine the stability of the target periodic orbit in the controlled system: 1) by directly studying the Floquet multipliers and 2) by use of the Hopf bifurcation theorem. We also propose an extension of the Pyragas control scheme for which the controlled system becomes a functional differential equation of neutral type. Using the observation that we are able to determine the direction of bifurcation by a relatively simple calculation of the root tendency, we find stability conditions for the periodic orbit as a solution of the neutral type equation.

Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients

In this article, we study the problem of estimating the pathwise Lyapunov exponent for linear stochastic systems with multiplicative noise and constant coefficients. We present a Lyapunov type matrix inequality that is closely related to this problem, and show under what conditions we can solve the matrix inequality. From this we can deduce an upper bound for the Lyapunov exponent. In the converse direction, it is shown that a necessary condition for the stochastic system to be pathwise asymptotically stable can be formulated in terms of controllability properties of the matrices involved.

Invariant measures for dichotomous stochastic differential equations in Hilbert spaces

Abstract We study existence of invariant measures for semilinear stochastic differential equations in Hilbert spaces. We consider infinite dimensional noise that is white in time and colored in space and we assume that the nonlinearities are Lipschitz continuous. We show that if the equation is dichotomous in the sense that the semigroup generated by the linear part is hyperbolic and the Lipschitz constants of the nonlinearities are not too large, then existence of a solution with bounded mean squares implies existence of an invariant measure. To cite this article: O. Van Gaans, S. Verduyn Lunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1083–1088.