Dirk Roose

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.

Wavelet-based image denoising using a Markov random field a priori model

This paper describes a new method for the suppression of noise in images via the wavelet transform. The method relies on two measures. The first is a classic measure of smoothness of the image and is based on an approximation of the local Holder exponent via the wavelet coefficients. The second, novel measure takes into account geometrical constraints, which are generally valid for natural images. The smoothness measure and the constraints are combined in a Bayesian probabilistic formulation, and are implemented as a Markov random field (MRF) image model. The manipulation of the wavelet coefficients is consequently based on the obtained probabilities. A comparison of quantitative and qualitative results for test images demonstrates the improved noise suppression performance with respect to previous wavelet-based image denoising methods.

An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions

This paper is concerned with the efficient computation of periodic orbits in large-scale dynamical systems that arise after spatial discretization of partial differential equations (PDEs). A hybrid Newton--Picard scheme based on the shooting method is derived, which in its simplest form is the recursive projection method (RPM) of Shroff and Keller [SIAM J. Numer. Anal., 30 (1993), pp. 1099--1120] and is used to compute and determine the stability of both stable and unstable periodic orbits. The number of time integrations needed to obtain a solution is shown to be determined only by the systems dynamics. This contrasts with traditional approaches based on Newtons method, for which the number of time integrations grows with the order of the spatial discretization. Two test examples are given to show the performance of the methods and to illustrate various theoretical points.

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.

ON PERIODIC ORBITS AND HOMOCLINIC BIFURCATIONS IN CHUA’S CIRCUIT WITH A SMOOTH NONLINEARITY

We present the bifurcation analysis of Chua’s circuit equations with a smooth nonlinearity, described by a cubic polynomial. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory. We emphasize how a combination of these approaches actually works in practice. We compare the dynamics of Chua’s circuit equations with piecewise-linear and with smooth nonlinearity. The dynamics of these two variants are similar, but we also present some differences. We conjecture that this similarity is due to the central role of homoclinicity in this model. We describe different ways in which the type of a homoclinic bifurcation influences the behavior of branches of periodic orbits. We present an overview of codimension 1 bifurcation diagrams for principal periodic orbits near homoclinicity for three-dimensional systems, both in the generic case and in the case of odd symmetry. Most of these diagrams actually occurs in the model. We found several homoclinic bifurcations of codimension 2, related to the so called resonant conditions. We study one of these bifurcations, a double neutral saddle loop.

Benchmarking the iPSC/2 hypercube multiprocessor

In this paper the performance of the Intel iPSC/2 hypercube multiprocessor is analyzed. Computation and communication performance for a number of benchmarks are presented. We derive some fundamental performance parameters of the machine. Further, we investigate the difference between several communication schemes. Using the results of our measurements we can highlight some features and peculiarities in the iPSC/2 hardware and software. Where possible we make a comparison with the iPSC/1 and Ncube hypercubes.

Detection of closed sharp edges in point clouds using normal estimation and graph theory

The reconstruction of a surface model from a point cloud is an important task in the reverse engineering of industrial parts. We aim at constructing a curve network on the point cloud that will define the border of the various surface patches. In this paper, we present an algorithm to extract closed sharp feature lines, which is necessary to create such a closed curve network. We use a first order segmentation to extract candidate feature points and process them as a graph to recover the sharp feature lines. To this end, a minimum spanning tree is constructed and afterwards a reconnection procedure closes the lines. The algorithm is fast and gives good results for real-world point sets from industrial applications.

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.

Numerical modelling of label-structured cell population growth using CFSE distribution data

BackgroundThe flow cytometry analysis of CFSE-labelled cells is currently one of the most informative experimental techniques for studying cell proliferation in immunology. The quantitative interpretation and understanding of such heterogenous cell population data requires the development of distributed parameter mathematical models and computational techniques for data assimilation.Methods and ResultsThe mathematical modelling of label-structured cell population dynamics leads to a hyperbolic partial differential equation in one space variable. The model contains fundamental parameters of cell turnover and label dilution that need to be estimated from the flow cytometry data on the kinetics of the CFSE label distribution. To this end a maximum likelihood approach is used. The Lax-Wendroff method is used to solve the corresponding initial-boundary value problem for the model equation. By fitting two original experimental data sets with the model we show its biological consistency and potential for quantitative characterization of the cell division and death rates, treated as continuous functions of the CFSE expression level.ConclusionOnce the initial distribution of the proliferating cell population with respect to the CFSE intensity is given, the distributed parameter modelling allows one to work directly with the histograms of the CFSE fluorescence without the need to specify the marker ranges. The label-structured model and the elaborated computational approach establish a quantitative basis for more informative interpretation of the flow cytometry CFSE systems.