Project B13 - Numerical Analysis for covariant Lyapunov vectors
Principal Investigator: Ingenuin Gasser, Reiner Lauterbach,
Background and Motivation
Dynamical systems play a fundamental role in mathematical theory and applications. However, due to high dimension and complexity it might be difficult and numerically expensive to explore whole systems. Nevertheless, local structures around simple object can be understood in terms of linear models. Well known results include the Hartman-Grobman theorem, describing the local flow around steady-states, and Floquet theory, which involves periodic solutions.
To treat more general solutions, Oseledets formulated his celebrated Multiplicative Ergodic Theorem in 1968. It ensures the existence of Lyapunov exponents and so-called covariant Lyapunov vectors (CLVs). Those quantities are a generalization to eigenvalues and eigenvectors of linearizations emerging from steady-states and periodic orbits. Thus, they can be used to determine physically relevant
directions for describing more complex objects.
Although Oseledets theorem has been available for quite some time, it was not until a few years ago that algorithms to compute CLVs emerged. One of the most popular approaches is the Ginelli algorithm from 2007. Since CLVs have such great physical importance, the algorithm was already tested in many different scenarios, yet rigorous mathematical analysis is still lacking.
Aims and Objectives
It is our task to verify previous applications and motivate further investigations by expanding the theoretical framework of CLV algorithms. The first step will be to construct a mathematically rigorous convergence proof of Ginelli’s algorithm in a fairly general setting. Further research might include different numerical attributes, specific classes of systems, and other algorithms.
PhD Student: Florian Noethen
Florian Noethen: A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors Physica D: Nonlinear Phenomena, in press, available online 1 March 2019.