# Computing covariant Lyapunov vectors in Hilbert spaces

## TL;DR

This paper extends the computation of covariant Lyapunov vectors to infinite-dimensional Hilbert spaces, providing a convergence theorem that relates the speed of convergence to spectral gaps, with potential applications in dynamical systems analysis.

## Contribution

It generalizes Ginelli's algorithm for CLV computation to Hilbert spaces and proves a convergence theorem based on spectral gaps, broadening applicability in infinite-dimensional systems.

## Key findings

- Convergence of the algorithm is related to spectral gaps between Lyapunov exponents.
- The proof relies on properties common to many versions of the multiplicative ergodic theorem.
- The approach applies to infinite-dimensional dynamical systems modeled in Hilbert spaces.

## Abstract

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 [Gonz\'alez-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.