Abstracts of invited speakers

Numerical methods for dynamical systems on unbounded domains

Joint work with Vera Thümmler.

The longtime behaviour of reaction diffusion systems

in on an unbounded domain may differ significantly from that on a bounded domain. For example, a travelling wave will persist on the infinite line while it will usually die out on any finite domain when it reaches the boundary. In this talk we present a method for freezing such solutions in a finite domain. We set up a differential algebraic PDE that may be used for time integration as well as for the bifurcation analysis of certain patterns. We will show how the method generalizes to infinite dimensional dynamical systems that are equivariant under the action of a Lie group. Advantages and disadvantages of such an approach are discussed for the case when one tries to prevent a plane spiral from rotating.

Numerical continuation for the Navier Stokes Equations

Set-oriented numerical methods in space mission design

Over the last years so-called set-oriented numerical methods have been developed for the reliable approximation of invariant objects of dynamical systems. With these techniques it is possible to approximate, for instance, invariant manifolds, invariant measures or almost invariant sets. In this talk an overview about the applicability of set oriented numerical tools in the context of space mission design will be given. The talk will particularly focus on

• the computation of invariant manifolds in connection with the NASA/JPL mission Genesis,
• the use of set oriented numerical tools for finding appropriate locations for formation flights in connection with the ESA mission Darwin, and
• the computation of transport phenomena in the solar system via the approximation of almost invariant sets for certain n-body problems.

Numerical challenges in the restricted three-body problem

Computation of periodic solution bifurcations in ODEs using bordered systems

Joint work with Sebius Doedel & Yuri Kuznetsov.

We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ordinary differential equations (ODEs), namely the fold, the period-doubling (or flip) bifurcation, and the torus (or Neimark-Sacker) bifurcation. In the fold and flip cases we append one scalar equation to the standard periodic boundary value problem (BVP) that defines the periodic solution; in the Neimark-Sacker case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of linear systems, whose sparsity structure (after discretization) is identical to that of the linearization of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software AUTO and COLSYS.

These methods are or will be incorporated in the new MATLAB software package MATCONT for which we thank A. M. Riet and W. Mestrom (Utrecht) and A. Dhooge (Gent).

Floquet theory as a computational tool

We describe how classical Floquet theory may be utilised to construct an efficient Fourier spectral algorithm for approximating periodic orbits in a continuation framework. If time permits, we shall also consider periodic connections, with points on the stable/unstable manifolds of periodic orbits being approximated by a Laguerre spectral method.

DDE-BIFTOOL, a software package for the bifurcation analysis of Delay Differential Equations

Joint work with Koen Engelborghs, Tatiana Luzyanina & Giovanni Samaey.

DDE-BIFTOOL is a Matlab-based software package for numerical bifurcation analysis of delay differential equations with fixed and/or state-dependent delays. The package contains procedures for stability analysis of steady state solutions of DDEs, computation of periodic solutions and their stability (using a collocation approach) and computation of homoclinic and heteroclinic orbits. We will illustrate the capabilities of DDE-BIFTOOL via the analysis of model problems (semiconductor lasers, epidemiology).

Defects in oscillatory media

Defects are modulated waves that connect two, possibly different, spatially-periodic travelling waves. Such waves have been observed in the CIMA reaction and in convection experiments. I will begin with a brief survey of different defects and their properties. Afterwards, I will focus on the interaction of a large standing pulse and small-amplitude plane waves. The main message of the talk is that defects can be investigated using homoclinic bifurcation theory in an appropriate infinite-dimensional setting.