Angular spectra of linear dynamical systems in discrete time
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916648779251712 |
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| author | Beyn, Wolf-Jürgen Hüls, Thorsten |
| author_facet | Beyn, Wolf-Jürgen Hüls, Thorsten |
| contents | In this work we introduce the notion of an angular spectrum for a linear discrete time nonautonomous dynamical system. The angular spectrum comprises all accumulation points of longtime averages formed by maximal principal angles between successive subspaces generated by the dynamical system. The angular spectrum is bounded by angular values which have previously been investigated by the authors. In this contribution we derive explicit formulas for the angular spectrum of some autonomous and specific nonautonomous systems. Based on a reduction principle we set up a numerical method for the general case; we investigate its convergence and apply the method to systems with a homoclinic orbit and a strange attractor. Our main theoretical result is a theorem on the invariance of the angular spectrum under summable perturbations of the given matrices (roughness theorem). It applies to systems with a so-called complete exponential dichotomy (CED), a concept which we introduce in this paper and which imposes more stringent conditions than those underlying the exponential dichotomy spectrum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_02694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Angular spectra of linear dynamical systems in discrete time Beyn, Wolf-Jürgen Hüls, Thorsten Dynamical Systems 37E45, 37M25, 34D09, 65Q10 In this work we introduce the notion of an angular spectrum for a linear discrete time nonautonomous dynamical system. The angular spectrum comprises all accumulation points of longtime averages formed by maximal principal angles between successive subspaces generated by the dynamical system. The angular spectrum is bounded by angular values which have previously been investigated by the authors. In this contribution we derive explicit formulas for the angular spectrum of some autonomous and specific nonautonomous systems. Based on a reduction principle we set up a numerical method for the general case; we investigate its convergence and apply the method to systems with a homoclinic orbit and a strange attractor. Our main theoretical result is a theorem on the invariance of the angular spectrum under summable perturbations of the given matrices (roughness theorem). It applies to systems with a so-called complete exponential dichotomy (CED), a concept which we introduce in this paper and which imposes more stringent conditions than those underlying the exponential dichotomy spectrum. |
| title | Angular spectra of linear dynamical systems in discrete time |
| topic | Dynamical Systems 37E45, 37M25, 34D09, 65Q10 |
| url | https://arxiv.org/abs/2404.02694 |