We will describe a numerical method to perform a bifurcation analysis of certain large-scale dynamical systems by putting an additional routine on top of an existing simulation code. A straightforward application of this idea is the computation of limit cycles by the single shooting method. However, the method can also be used to compute steady-state solutions. In fact, one could even consider to use our algorithm to study systems with unknown equations, as long as it is possible to construct a time integrator for the system. (This is done in the presentation of Prof. Dr. Y. Kevrekidis.) Even the timestepper is not essential: one can also use the method to perform a bifurcation analysis of a high-dimensional map. The main assumption made is that the Jacobian of the map - the monodromy matrix for periodic solutions - has only few large eigenvalues (where large means: close to or outside the unit circle in the complex plane.)

All the above problems have one thing in common: they can be written as a large nonlinear system of equations with one or more parameters. After linearization, we obtain a bordered matrix with the Jacobian of the map minus the identity matrix as the big upper left block, and rows and columns corresponding to additional constraints such as a phase- or pseudo arclength condition, and the parameters respectively. In this presentation, we will present a method, inspired on the Recursive Projection Method of Shroff and Keller, to solve these nonlinear systems efficiently, simultaneously computing the dominant eigenvalues of the Jacobian of the map.

We will also present some results for steady-state and periodic solutions of PDE systems.