Angular spectra of linear dynamical systems in discrete time

Wolf-Jürgen Beyn; Thorsten Hüls

doi:10.3934/dcdsb.2024144

Article Contents

2025, Volume 30, Issue 6: 1859-1902. Doi: 10.3934/dcdsb.2024144

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Angular spectra of linear dynamical systems in discrete time

Wolf-Jürgen Beyn and
Thorsten Hüls^,

Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany

^*Corresponding author: Thorsten Hüls
^*Corresponding author: Thorsten Hüls

Received: April 2024

Revised: September 2024

Early access: October 16, 2024

Published online: October 16, 2024

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Abstract

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.

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Mathematics Subject Classification: Primary: 37E45, 37M25, 34D09, 65Q10.

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Figure 1. Computation of the angle between two planes in $ \mathbb{R}^3 $

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Figure 2. Outer angular spectrum of the autonomous system $ u_{n+1} = A(\rho, \varphi)u_n $ for $ A(\rho, \varphi) $ from (10) as a function of the parameters $ \rho \in (0, 1] $, $ \varphi\in (0, \frac{\pi}{2}] $

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Figure 3. Illustration of spectral intervals (red), of resolvent intervals (blue) and the construction of spectral bundles with corresponding fiber projectors

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Figure 4. Computation of the second angular spectrum for the non-resonant case $ \varphi = 1.25 $. Left panel: For each $ (\gamma_w, \rho) $-pair we obtain point spectrum only (red plane at $ 0 $ and blue surface). Right panel: Point spectrum for fixed $ \rho = 0.2 $

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Figure 5. Computation of the second angular spectrum for the resonant case $ \varphi = \frac 25 \pi $. Left panel: For each $ (\gamma_w, \rho) $-pair the spectrum consists of the point spectrum $ 0 $ (red plane) and of the interval between the orange and the blue surface. Note that in certain areas, both surfaces coincide and the spectral intervals degenerate to a point. In the intersection of this figure with the plane at $ \rho = 0.2 $ (right panel) the spectrum consists of the curves and the grey area between them

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Figure 6. Center part of a homoclinic orbit of (61) (left). The same orbit in phase space (right) with approximations of unstable (red) and stable manifolds of $ \xi $

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Figure 7. Distance of a homoclinic orbit¹ $ (\bar x_n)_{n\in\{0, 1, \dots, 2000\}} $ of the $ 3 $D-Hénon system (61) to the fixed point $ \xi $ in a semilogarithmic scale

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Figure 8. Construction of multi-humped orbits

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Figure 9. Part of an $ F_{0.05} $-orbit $ (\bar x_n)_{n\in\{2300, \dots, 2500\}} $ (connected with lines) of the Lorenz system (80). For $ n = 2400 $ the three trace spaces $ \mathcal{W}_n^{1, 2, 3} $ are shown

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Table 1. Outer angular values for (9)

angular value	achieved in
$ \theta_1^{\inf, \varliminf} = 0 $	$ V = \mathcal{W}_0^1 $
$ \theta_1^{\sup, \varlimsup} = \varphi $	$ V= {\mathrm{span}} \begin{pmatrix}1&0&0\end{pmatrix}^\top $
$ \theta_2^{\inf, \varliminf} = 0 $	$ V = \mathcal{W}_0^2 $
$ \theta_2^{\sup, \varlimsup} = \varphi $	$ V= {\mathrm{span}} \left(\begin{pmatrix}0&0&1\end{pmatrix}^\top, \begin{pmatrix}1&0&0\end{pmatrix}^\top\right) $

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Table 2. Construction of $ (A_n)_{n\in \mathbb{N}_0} $

$n$	0	1	2	3	4	5	6	7	8	$ \dots$	15	16	17	$ \dots$	32	33
$A_n$	$D$	$D$	$X$	$D$	$D$	$D$	$D$	$X$	$D$	$\dots$	$D$	$X$	$D$	$ \dots$	$D$	$X$

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Table 3. Numerical computation of the dichotomy spectrum and of outer angular spectra for the nonautonomous $ 3 $D-Hénon system (62)

$ N $	$ \Sigma_{\mathrm{ED}}^{\mathrm{approx}} $	$ \Sigma_1^N $	$ \Sigma_2^N $
$ 50 $	$ [0.775, 0.787]\cup[1.467, 1.482] $	$ \{0.358\}\cup [1.108, 1,264] $	$ \{0.487\}\cup[1.211, 1.275] $
$ 100 $	$ [0.776, 0.785]\cup[1.468, 1.482] $	$ \{0.178\}\cup [1.222, 1,307] $	$ \{0.242\}\cup[1.289, 1.296] $
$ 1000 $	$ [0.779, 0.784]\cup[1.470, 1.478] $	$ \{0.018\}\cup [1.325, 1,333] $	$ \{0.024\}\cup[1.324, 1.325] $
$ 2000 $	$ [0.779, 0.784]\cup[1.471, 1.477] $	$ \{0.009\}\cup [1.330, 1,331] $	$ \{0.012\}\cup[1.326, 1.327] $

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Table 4. Numerical computation of the dichotomy spectrum and of outer angular spectra for the autonomous $ 3 $D-Hénon system (63)

$ N $	$ \Sigma_{\mathrm{ED}}^{\mathrm{approx}} $	$ \Sigma_1^N $	$ \Sigma_2^N $
$ 50 $	$ [0.781, 0.782]\cup[1.473, 1.474] $	$ \{10^{-16}\}\cup [1.328, 1,344] $	$ \{10^{-16}\}\cup[1.320, 1.337] $
$ 100 $	$ [0.781, 0.782]\cup[1.473, 1.474] $	$ \{10^{-16}\}\cup [1.328, 1,341] $	$ \{10^{-16}\}\cup[1.321, 1.334] $
$ 1000 $	$ [0.781, 0.782]\cup[1.473, 1.474] $	$ \{10^{-16}\}\cup [1.335, 1,336] $	$ \{10^{-15}\}\cup[1.328, 1.328] $
$ 2000 $	$ [0.781, 0.782]\cup[1.473, 1.474] $	$ \{10^{-16}\}\cup [1.335, 1,336] $	$ \{10^{-15}\}\cup[1.328, 1.328] $

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Table 5. Numerical computation of the dichotomy spectrum and of outer angular spectra for the $ 3 $D-Hénon system w.r.t. multi-humped homoclinic orbits with center parts of length $ M $

$ M $	$ \Sigma_{\mathrm{ED}}^{\mathrm{approx}} $
$ 50 $	$ [0.7230, 0.7234]\cup [0.8298, 0.8302]\cup[1.4992, 1.4994] $
$ 100 $	$ [0.7492, 0.7494]\cup[0.8077, 0.8079]\cup[1.4868, 1.4870] $
$ 200 $	$ [0.7634, 0.7662] \cup [0.7935, 0.7966]\cup [1.4775, 1.4827] $
$ 400 $	$ [0.7733, 0.7884] \cup [1.4741, 1.4791] $
$M$	$\Sigma_1^{2001}$	$\Sigma_2^{2001}$
$50$	$\{0.353, 1.135, 1.260\} $	$\{0.480, 1.248, 1.264\}$
$100$	$\{0.177, 1.229, 1.300\}$	$\{0.239, 1.295, 1.296\}$
$200$	$\{0.088, 1.281, 1.318\}$	$\{0.120, 1.312, 1.313\}$
$400$	$\{0.044\}\cup[1.311, 1.314]$	$\{0.060\}\cup[1.320, 1.321]$

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Table 6. Dichotomy spectrum and angular spectra for the $ h $-step Lorenz map w.r.t. an orbit on the Lorenz attractor

$ h $	$ \Sigma_{\mathrm{ED}}^{\mathrm{approx}} $
$ 0.05 $	$ [0.4821, 0.4833]\cup [0.9995, 1.0005]\cup[1.0445, 1.0478] $
$ 0.1 $	$ [0.2325, 0.2332]\cup [0.9995, 1.0007]\cup[1.0928, 1.0971] $
$ 0.2 $	$ [0.0542, 0.0544]\cup [0.9994, 1.0006]\cup[1.1956, 1.2004] $
$h$	$\Sigma_1^{10001}$	$\Sigma_2^{10001}$
$0.05$	$\{0.2039, 0.3803, 0.4234\} $	$\{0.0689, 0.3604, 0.4188\}$
$0.1$	$\{0.3925, 0.7282, 0.8268\} $	$\{0.1356, 0.7021, 0.8197\}$
$0.2$	$\{0.6552, 0.7334, 0.9727\} $	$\{0.2475, 0.7934, 0.9752\}$

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Table 7. Angle between successive iterates on average for the $ h $-step Lorenz map (80)

$ h $	$ 0.05 $	$ 0.1 $	$ 0.2 $
angle on average	$ 0.4227 $	$ 0.8322 $	$ 1.0993 $

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Table 8. Normalized angular values $ \theta_{1, h, T}^{\mathrm{cont}} $ and $ \theta_{2, h, T}^{\mathrm{cont}} $ for the $ h $-step Lorenz map (80)

$ h $	$ 0.025 $	$ 0.05 $	$ 0.1 $	$ 0.2 $
$ \theta_{1, h, T}^{\mathrm{cont}} $	$ 8.4798 $	$ 8.4672 $	$ 8.2896 $	$ 4.8752 $
$ \theta_{2, h, T}^{\mathrm{cont}} $	$ 8.3816 $	$ 8.3753 $	$ 8.2209 $	$ 4.8941 $

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References

[1]	L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	B. Aulbach and J. Kalkbrenner, Exponential forward splitting for noninvertible difference equations, Comput. Math. Appl., 42 (2001), 743-754. doi: 10.1016/S0898-1221(01)00194-8.
[3]	B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Differ. Equations Appl., 7 (2001), 895-913. doi: 10.1080/10236190108808310.
[4]	A. Babiarz, A. Czornik, M. Niezabitowski, E. Barabanov and A. Vaidzelevich and A. Konyukh, Relations between Bohl and general exponents, Discrete Contin. Dyn. Syst., 37 (2017), 5319-5335. doi: 10.3934/dcds.2017231.
[5]	T. Bendokat, R. Zimmermann and P.-A. Absil, A Grassmann manifold handbook: Basic geometry and computational aspects, Adv. Comput. Math., 50 (2024), Paper No. 6, 51 pp. doi: 10.1007/s10444-023-10090-8.
[6]	W.-J. Beyn, G. Froyland and T. Hüls, Angular values of nonautonomous and random linear dynamical systems: Part Ⅰ—Fundamentals, SIAM J. Appl. Dyn. Syst., 21 (2022), 1245-1286. doi: 10.1137/20M1387730.
[7]	W.-J. Beyn and T. Hüls., Error estimates for approximating non-hyperbolic heteroclinic orbits of maps, Numer. Math., 99 (2004), 289-323. doi: 10.1007/s00211-004-0563-4.
[8]	W.-J. Beyn and T. Hüls, Angular values of nonautonomous linear dynamical systems: Part Ⅱ – Reduction theory and algorithm, SIAM J. Appl. Dyn. Syst., 22 (2023), 162-198. doi: 10.1137/20M1387766.
[9]	W.-J. Beyn and T. Hüls, On the smoothness of principal angles between subspaces and their application to angular values of dynamical systems, Dyn. Syst., 39 (2024), 461-499. doi: 10.1080/14689367.2024.2322156.
[10]	W.-J. Beyn, T. Hüls and A. Schenke, Symbolic coding for noninvertible systems: Uniform approximation and numerical computation, Nonlinearity, 29 (2016), 3346-3384. doi: 10.1088/0951-7715/29/11/3346.
[11]	W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629. Springer, Berlin, 1978. doi: 10.1007/BFb0067780.
[12]	J. L. Daleckiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, American Mathematical Society, Providence, R.I., 1974.
[13]	G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, fourth edition, 2013.
[14]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981. doi: 10.1007/BFb0089647.
[15]	T. Hüls, Numerische Approximation Nicht-Hyperbolischer Heterokliner Orbits, Shaker-Verlag, Aachen, PhD thesis, Bielefeld University, 2003.
[16]	T. Hüls, A contour algorithm for computing stable fiber bundles of nonautonomous, noninvertible maps, SIAM J. Appl. Dyn. Syst., 15 (2016), 923-951. doi: 10.1137/140999815.
[17]	T. Hüls, Computing stable hierarchies of fiber bundles, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3341-3367. doi: 10.3934/dcdsb.2017140.
[18]	J. Kalkbrenner, Exponentielle Dichotomie und Chaotische Dynamik Nichtinvertierbarer Differenzengleichungen, volume 1 of Augsburger Mathematisch-Naturwissenschaftliche Schriften. Dr. Bernd Wißner, Augsburg, 1994.
[19]	A. B. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge, 1995.
[20]	E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
[21]	K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics reported, Dynam. Report. Ser. Dynam. Systems Appl., 1 (1988), 265-306. doi: 10.1007/978-3-322-96656-8_5.
[22]	O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.
[23]	K. Petersen, Ergodic Theory, volume 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989.
[24]	C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Operator Theory, 73 (2012), 107-151. doi: 10.1007/s00020-012-1959-7.
[25]	C. Pötzsche, Continuity of the Sacker-Sell spectrum on the half line, Dyn. Syst., 33 (2018), 27-53. doi: 10.1080/14689367.2017.1293613.
[26]	C. Pötzsche and E. Russ, Continuity and invariance of the Sacker-Sell spectrum, J. Dynam. Differential Equations, 28 (2016), 533-566. doi: 10.1007/s10884-015-9515-1.
[27]	R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.