Matrix-Free Numerical Torus Bifurcation of Periodic Orbits
Kurt Georg
Colorado State University

We consider systems

$$\dot u = f(u,\lambda)$$

where $f:{\cal R}^n\times{\cal R} \rightarrow {\cal R}^n$. Such systems often arise from space discretizations of parabolic PDEs. We are interested in branches (with respect to $\lambda$) of periodic solutions of such systems.

In the present paper we describe a numerical continuation method for tracing such branches. Our methods are matrix-free, i.e., Jacobians are only implemented as actions, this enables us to allow for large $n$. Of particular interest is the detection and precise numerical approximation of bifurcation points along such branches: especially period-doubling and torus bifurcation points. This will also be done in a matrix-free context combining Arnoldi iterations (to obtain coarse information) with the calculation of suitable test functions (for precise approximations). We illustrate the method with the one- and two-dimensional Brusselator.

Collaborators: Eugene Allgower, Ulf Garbotz