# American Institute of Mathematical Sciences

July  2021, 8(3): 325-352. doi: 10.3934/jcd.2021014

## Computing Covariant Lyapunov Vectors in Hilbert spaces

 Department of Mathematics, Bundesstr. 55, 20146 Hamburg, Germany

* Corresponding author: Florian Noethen

Received  August 2020 Published  July 2021 Early access  July 2021

Fund Project: The author is supported by DFG grant 274762653

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 [19]. 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.

Citation: Florian Noethen. Computing Covariant Lyapunov Vectors in Hilbert spaces. Journal of Computational Dynamics, 2021, 8 (3) : 325-352. doi: 10.3934/jcd.2021014
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Ginelli's algorithm at the level of Grassmannians. The algorithm approximates CLVs (or Oseledets spaces) by pushing forward and backward linear perturbations along the $\sigma$-trajectory of a state $\omega$ via the linear propagator $\mathcal{L}$. First, a randomly chosen subspace $W$ of dimension $d_1+\dots+d_i$ is propagated from the past to the future. Then, a new randomly chosen subspace $\tilde{W}$ of dimension $d_i$ inside the forward propagated subspace of $W$ is propagated backward from the future to the present to provide an approximation of the $i^{\text{th}}$ Oseledets space at $\omega$. Here, $d_1, \dots, d_i$ are the multiplicities of the first $i$ Lyapunov exponents
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