### Angular values of nonautonomous and random linear dynamical systems

Beyn W-J, Froyland G, Hüls T (2020)
arXiv:2012.11305.

Preprint | Englisch

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Abstract / Bemerkung
We introduce the notion of angular values for nonautonomous linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\frac{\pi}{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. We treat angular values in the nontrivial autonomous case in detail and show how angular values may be obtained from generalized eigenspaces. This leads to a numerical algorithm for computing angular values via Schur decompositions.
Erscheinungsjahr
2020
Zeitschriftentitel
arXiv:2012.11305
Page URI
https://pub.uni-bielefeld.de/record/2955504

### Zitieren

Beyn W-J, Froyland G, Hüls T. Angular values of nonautonomous and random linear dynamical systems. arXiv:2012.11305. 2020.
Beyn, W. - J., Froyland, G., & Hüls, T. (2020). Angular values of nonautonomous and random linear dynamical systems. arXiv:2012.11305
Beyn, W. - J., Froyland, G., and Hüls, T. (2020). Angular values of nonautonomous and random linear dynamical systems. arXiv:2012.11305.
Beyn, W.-J., Froyland, G., & Hüls, T., 2020. Angular values of nonautonomous and random linear dynamical systems. arXiv:2012.11305.
W.-J. Beyn, G. Froyland, and T. Hüls, “Angular values of nonautonomous and random linear dynamical systems”, arXiv:2012.11305, 2020.
Beyn, W.-J., Froyland, G., Hüls, T.: Angular values of nonautonomous and random linear dynamical systems. arXiv:2012.11305. (2020).
Beyn, Wolf-Jürgen, Froyland, Gary, and Hüls, Thorsten. “Angular values of nonautonomous and random linear dynamical systems”. arXiv:2012.11305 (2020).

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### Quellen

arXiv: 2012.11305