Koen Engelborghs

Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL

We describe DDE-BIFTOOL, a Matlab package for numerical bifurcation analysis of systems of delay differential equations with several fixed, discrete delays. The package implements continuation of steady state solutions and periodic solutions and their stability analysis. It also computes and continues steady state fold and Hopf bifurcations and, from the latter, it can switch to the emanating branch of periodic solutions. We describe the numerical methods upon which the package is based and illustrate its usage and capabilities through analysing three examples: two models of coupled neurons with delayed feedback and a model of two oscillators coupled with delay.

Limitations of a class of stabilization methods for delay systems

We investigate limitations of certain stabilization methods for time-delay systems. The class of methods under consideration implements the control law through a Volterra integral equation of the second kind. Using as an example the pole placement approach of Manitius and Olbrot (1979), we illustrate how instability of the difference part of the control law leads to instability in the closed-loop system, in the case that implementation is done via numerical quadrature. The outcome of our analysis provides computable limitations to stability and a maximum allowable size of the (input) delay.

Numerical bifurcation analysis of delay differential equations

Numerical methods for the bifurcation analysis of delay differential equations (DDEs) have only recently received much attention, partially because the theory of DDEs (smoothness, boundedness, stability of solutions) is more complicated and less established than the corresponding theory of ordinary differential equations. As a consequence, no established software packages exist at present for the bifurcation analysis of DDEs. We outline existing numerical methods for the computation and stability analysis of steady-state solutions and periodic solutions of systems of DDEs with several constant delays.

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points.

Sensitivity to Infinitesimal Delays in Neutral Equations

The stability of a steady state solution of a neutral functional differential equation can be sensitive to infinitesimal changes in the delays. This phenomenon is caused by the behavior of the essential spectrum and is determined by the roots of an exponential polynomial. Avellar and Hale [J. Math. Anal. Appl., 73 (1980), pp. 434--452] have considered the case of multiple fixed and nonzero delays. In the first part of this paper their results are illustrated by means of spectral plots. In the second part we extend the theory of Avellar and Hale to the limit case whereby some of the delays are brought to zero, which may lead to characteristic roots with arbitrarily large real part. Necessary and sufficient conditions are provided. Using these results we show that the ratio of the delays plays a crucial role when several delays tend to zero simultaneously. As an illustration of the theory, we analyze the robustness of a boundary controlled PDE in the presence of a small feedback delay.

Computation, Continuation and Bifurcation Analysis of Periodic Solutions of Delay Differential Equations

We present a new numerical method for the ecient computation of periodic solutions of nonlinear systems of Delay Dierential Equations (DDEs) with several discrete delays. This method exploits the typical spectral properties of the monodromy matrix of a DDE and allows eective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. We show that the method is particularly suited to trace a branch of periodic solutions using continuation and can be used to locate bifurcation points with good accuracy.

Computing Stability of Differential Equations with Bounded Distributed Delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation

NUMERICAL BIFURCATION ANALYSIS OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

\dot y(t) = a(t)y(t)+b(t)y(t-\tau) + f(t)

Numerical Computation of Connecting Orbits in Delay Differential Equations

by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given.