Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Computing stable hierarchies of fiber bundles

Pages: 3341 - 3367, Volume 22, Issue 9, November 2017      doi:10.3934/dcdsb.2017140

       Abstract        References        Full Text (4358.7K)       Related Articles       

Thorsten Hüls - Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld, Germany (email)

Abstract: 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 [38,39]. 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.

Keywords:  Hierarchy of invariant fiber bundles, dynamical systems, noninvertible and nonautonomous dynamics, numerical approximation, contour algorithm.
Mathematics Subject Classification:  Primary: 37B55, 37D10; Secondary: 65P40.

Received: September 2016;      Revised: December 2016;      Available Online: April 2017.