[show abstract][hide abstract] ABSTRACT: We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.
[show abstract][hide abstract] ABSTRACT: In this paper we investigate the sensitivity of the stability of neutralfunctional differential equations with respect to changes in thedelays. This sensitivity is caused by the behaviour of the essentialspectrum which, in turn, is determined by the roots of an exponentialpolynomial. In , Avellar and Hale considered the case ofmultiple fixed and nonzero delays. In a first part of this paper wevisualize and interpret their results by means of computed spectralplots. In a second part we ...
SIAM J. Control and Optimization. 01/2002; 40:1134-1158.
[show abstract][hide abstract] ABSTRACT: Floquet multipliers determine the local asymptotic stability of
a periodic solution and, in the context of parameter dependence,
determine also its bifurcations.
This paper deals with numerical aspects of the computation of
the Floquet multipliers for three classes of
functional differential equations: ordinary differential equations (ODEs),
differential equations with constant delay (DDEs) and differential
equations with state-dependent delay (sd-DDEs).
Using a collocation approach for computing periodic solutions,
we obtain an approximation of
the (corresponding) monodromy operator, a
monodromy matrix. The eigenvalues of this matrix form an
approximation to the Floquet multipliers.
The accuracy of the computed multipliers is an important issue
in bifurcation analysis of a dynamical system.
As far as we know, no prior work on the study of
the convergence and accuracy of computed Floquet multipliers
for DDEs and sd-DDEs exists.
We analyse the dependency of the accuracy of the computed
multipliers on the parameters and on the type of collocation
In particular, we show that the accuracy of the computed trivial multiplier
is not always comparable to the accuracy of the computed periodic solution
and the accuracy of the other computed multipliers.
International Journal of Bifurcation and Chaos 01/2002; 12:2977-2989. · 0.92 Impact Factor
[show abstract][hide abstract] ABSTRACT: 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.
Journal of Computational and Applied Mathematics 08/2000; 125:265-275. · 0.99 Impact Factor
[show abstract][hide abstract] ABSTRACT: 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.
Advances in Computational Mathematics 04/1999; 10(3):271-289. · 1.47 Impact Factor
[show abstract][hide abstract] ABSTRACT: We present a new numerical method for the ecient computation of periodic solutions of nonlin- ear 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 solu- tion. 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.
International Journal of Bifurcation and Chaos 08/1997; 7(11):2547-2560. · 0.92 Impact Factor
[show abstract][hide abstract] ABSTRACT: DDE-BIFTOOL is a Matlab software package for the stability and bifurcation analysis of parameter-dependent systems of delay
differential equations. Using continuation, branches of steady state solutions and periodic solutions can be computed. The
local stability of a solution is determined by computing relevant eigenvalues (steady state solutions) or Floquet Multipliers
(periodic solutions). Along branches of solutions, bifurcations can be detected and branches of fold or Hopf bifurcation points
can be computcd. We outline the capabilities of the package and some of the numerical methods upon which it is based. We illustrate
the usage of the package for the analysis of two model problems and we outline applications and extensions towards controller
synthesis problems. We explain how its stability routines can be used for the implementation of the continuous pole placement
method, which allows to solve stabilization problems where multiple parameters need to be tuned simultaneously.