The Karhunen Loeve Decomposition with Symmetry

Michael Dellnitz, Universitaet Bayreuth, Mathematisches Institut

The Karhunen-Lo\`eve decomposition (KL-decomposition) -- also known as the proper orthogonal decomposition (POD) -- is now a well established tool for the numerical investigation of nonlinear phenomena in dynamical systems. Since dynamical systems frequently possess symmetries it is natural to ask how well the Karhunen-Lo\`eve decomposition works for such systems. Motivated by this question we investigate the symmetry properties of the KL-decomposition and we show, for instance, under which circumstances symmetries are inherited by the method in an appropriate way. Simultaneously it turns out that the KL-decomposition can be used as a tool to detect transitions in symmetry properties of complicated (e.g.\ chaotic) dynamical motion.