Matrix-free numerical torus bifurcation of periodic orbits

We consider systems φ̇ = f(φ, λ) where f : R×R → R. Such systems often arise from space discretizations of parabolic PDEs. We are interested in branches (with respect to λ) 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 oneand two-dimensional Brusselator. 1 – Introduction Recently, Georg [5] discussed a general setting for performing numerical continuation in a matrix-free setting. Transpose-free iterative linear solvers (see, e.g., [15]) can be effectively incorporated into such large-scale problems. A frequent application of numerical continuation concerns the detection of singularities and bifurcation points on a solution branch. By means of suitable test