Computer demonstrations

Annick Dhooge, University of Gent, Belgium

MATCONT: a MATLAB package for numerical bifurcation analysis of ODE's

MATCONT is a graphical MATLAB package for the interactive numerical study of dynamical systems. It is developed in parallel with the continuation toolbox CL_MATCONT. Both MATCONT and CL_MATCONT allow to compute curves of equilibria, limit points, Hopf points, limit cycles and period doubling bifurcation points of limit cycles. MATCONT makes the MATLAB odesuite for time integration interactively available and can use the MATLAB Symbolic Toolbox for computing derivatives whenever it is installed. In handling limit cycles and their bifurcations it uses the MATLAB sparse matrix routines to exploit the sparsity of the resulting linear systems. We present the use of MATCONT in a few interesting test examples.

Gabriel Lord, Heriot Watt University, Edinburgh

Numerical computation of travelling waves in stochastic pdes

Joint work with Jacques Rougemont.

We show numerical computations of stochastic pdes with stochastic forcing. We are particular interested noise that is white noise in time and spatially in a Gevrey class of regularity. Numerical computations confirm analysis that our numerical scheme preserves the regularity (and in fact an improved error estimate is also obtained).

We will show an example of Gevrey noise and some sample computations for Ginzburg-Landau and FitzHugh-Nagumo type equations.

Tatiana Luzyanina & Giovanni Samaey, Katholieke Universiteit Leuven, Belgium

Using DDE-BIFTOOL for the bifurcation analysis of Delay Differential Equations: a demonstration.

DDE-BIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with several constant and state-dependent delays. The package allows to compute and continue steady state and periodic solutions and to find steady state fold and Hopf bifurcations. One can also switch, from the latter, to an emanating branch of periodic solutions. Homoclinic and heteroclinic orbits can also be computed, given a sufficiently accurate time profile of the solution. The package allows to analyse the stability of steady state solutions by approximating the rightmost, stability-determining roots of the characteristic equation. For periodic solutions, approximations to the largest Floquet multipliers are computed. We illustrate the use of DDE-BIFTOOL through a step-by-step analysis of a demo system.

Bart Oldeman, University of Bristol

Homoclinic branch switching with AUTO2000

AUTO2000 is a new version of the well-known continuation software AUTO, Among other things it features an improved user interface and graphics. Scripts written in the Python programming language can be used to steer the calculations. These new capabilities were primarily added by Randy Paffenroth at Caltech.

As part of my PhD project I implemented a homoclinic branch switching method in AUTO, to be able to numerically switch from a 1-homoclinic orbit to an n-homoclinic orbit for any n. The demonstration features both AUTO2000 and the homoclinic branch switching method.

Gábor Orosz, Budapest University of Technology and Economics, Hungary

Bifurcation analysis and numerical simulation of a car-following system

Joint work with Gábor Stépán.

A simple car-following model with time delay in its control is presented for cars running along a ring. The linear stability of this system is investigated, and explicit stability boundaries are determined for the case the car-drivers have the same control characteristics. We study the non-linear behaviour of the two- and three-car systems. With the help of a new computer code, we check all the analytical results and also visualise the motion of the system. With the help of the simulation program, the nonlinear oscillations and also travelling stop-and-go traffic jams are observable.

Reza Rokni, University of Exeter

SSMan1D: One-dimensional strong (un)stable manifold computations

Joint work with Hinke Osinga & Stuart Townley.

SSMan1D is a package for use in the DsTool environment. It is designed for the computation of one-dimensional strong stable or unstable manifolds of non-hyperbolic equilibria of vector fields, but it also works for hyperbolic equilibria. SSMan1D computes the one-dimensional manifold associated with the strongest stable or unstable eigenvalue of an equilibrium, provided this eigenvalue is real and unique.

Frank Schilder, Universität Ilmenau, Germany

TorCont & The Math-Template-Library

Currently, we are developing a library (the Math-Template- Library or mtl for short), which is especially designed to handle parameter-dependent nonlinear zero-problems of almost arbitrary type. This library is based on new and powerful paradigms for developing numerical software or doing numerical computations.

Examples of use are the construction of continuation codes for fixed points and periodic and quasiperiodic solutions of ODEs. Especially in the case of quasiperiodic solutions the abilities of the mtl-programming style become visible.

Together with the software demo we present our approach for the computation and continuation of quasiperiodic solutions of ODEs. In addition, we briefely describe ideas for stability- and bifurcation analysis.