Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project B3

Numerical Analysis of equivariant evolution equations

16-037 Thorsten Hüls.
Computing stable hierarchies of fiber bundles


Stable fiber bundles are important structures for understanding nonautonomous dynamics. These sets have a hierarchical structure ranging from stable to strong stable fibers. First, we compute corresponding structures for linear systems and prove an error estimate. The spectral concept of choice is the Sacker-Sell spectrum that is based on exponential dichotomies. Secondly, we tackle the nonlinear case and propose an algorithm for the numerical approximation of stable hierarchies in nonautonomous difference equations. This method generalizes the contour algorithm for computing stable fibers from [Hüls, 2016]. It is based on Hadamard's graph transform and approximates fibers of the hierarchy by zero-contours of specific operators. We calculate fiber bundles and illustrate errors involved for several examples, including a nonautonomous Lorenz model.