arXiv subjects

Florian Noethen

Publications and source records attributed to Florian Noethen.

4 recordsLinked to original sources

arXiv2024

Strong Fast Invertibility and Lyapunov Exponents for Linear Systems

In 2019 Anthony Quas, Philippe Thieullen and Mohamed Zarrabi introduced the concept of strong fast invertibility for linear cocycles. It relates the growth of volumes between different initial times and, together with a condition on singular value gaps, yields the existence of a dominated splitting of the dynamics. The properties of this splitting largely coincide with those for systems with stable Lyapunov exponents. In this article, we take a closer look at strongly fast invertible systems with bounded coefficients. By linking the dimensions at which a system admits strong fast invertibility to the multiplicities of Lyapunov exponents, we are able to give a full characterization of regular strongly fast invertible systems similar to that of systems with stable Lyapunov exponents. In particular, we show that the stability of Lyapunov exponents implies strong fast invertibility (even in the absence of regularity). Central to our arguments are certain induced systems on spaces of exterior products that represent the evolution of volumes. Finally, we derive convergence results for the computation of Lyapunov exponents via Benettin's algorithm using perturbation theory. While the stronger assumption of stable Lyapunov exponents clearly leaves more freedom on how to choose stepsizes, we derive conditions for the stepsizes with which convergence can be ensured even if a system is only strongly fast invertible.

math.DS↗

arXiv2019

Computing covariant Lyapunov vectors in Hilbert spaces

Covariant Lyapunov vectors (CLVs) are intrinsic modes that describe long-term linear perturbations of solutions of dynamical systems. With recent advances in the context of semi-invertible multiplicative ergodic theorems, existence of CLVs has been proved for various infinite-dimensional scenarios. Possible applications include the derivation of coherent structures via transfer operators or the stability analysis of linear perturbations in models of increasingly higher resolutions. We generalize the concept of Ginelli's algorithm to compute CLVs in Hilbert spaces. Our main result is a convergence theorem in the setting of [González-Tokman, C. and Quas, A., A semi-invertible operator Oseledets theorem, Ergodic Theory and Dynamical Systems, 34.4 (2014), pp. 1230-1272]. The theorem relates the speed of convergence to the spectral gap between Lyapunov exponents. While the theorem is restricted to the above setting, our proof requires only basic properties that are given in many other versions of the multiplicative ergodic theorem.

math.DS↗

arXiv2019

Well-separating common complements of a sequence of subspaces of the same codimension in a Hilbert space are generic

Given a family of subspaces we investigate existence, quantity and quality of common complements in Hilbert spaces and Banach spaces. In particular we are interested in complements for countable families of closed subspaces of finite codimension. Those families naturally appear in the context of exponential type splittings like the multiplicative ergodic theorem, which recently has been proved in various infinite-dimensional settings. In view of these splittings, we show that common complements with subexponential decay of quality are generic in Hilbert spaces. Moreover, we prove that the existence of one such complement in a Banach space already implies that they are generic.

math.FA↗

arXiv2019

A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors

Linear perturbations of solutions of dynamical systems exhibit different asymptotic growth rates, which are naturally characterized by so-called covariant Lyapunov vectors (CLVs). Due to an increased interest of CLVs in applications, several algorithms were developed to compute them. The Ginelli algorithm is among the most commonly used. Although several properties of the algorithm have been analyzed, there exists no mathematically rigorous convergence proof yet. In this article we extend existing approaches in order to construct a projector-based convergence proof of Ginelli's algorithm. One of the main ingredients will be an asymptotic characterization of CLVs via the Multiplicative Ergodic Theorem. In the proof, we keep a rather general setting allowing even for degenerate Lyapunov spectra.

math.DS↗