Matrix-Free Numerical Torus Bifurcation of Periodic Orbits

E. L. Allgower, U. Garbotz and K. Georg

Colorado State University

We describe a numerical continuation method for tracing branches of periodic solutions of dynamical systems in a matrix-free context i.e., Jacobians are only implemented as actions. This enables us to handle large systems, such as those arising from discretization of PDEs. Of particular interest is the detection and precise numerical approximation of bifurcation points along 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.