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
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dyn. Syst., 13 (2005), 469-490.  doi: 10.3934/dcds.2005.13.469.  Google Scholar

[3]

C. Blachut and C. González-Tokman, A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems, J. Comput. Dyn., 7 (2020), 369-399.  doi: 10.3934/jcd.2020015.  Google Scholar

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A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2377-2403.  doi: 10.3934/dcds.2016.36.2377.  Google Scholar

[5]

M. CarluF. GinelliV. Lucarini and A. Politi, Lyapunov analysis of multiscale dynamics: The slow bundle of the two-scale Lorenz 96 model, Nonlin. Processes Geophys., 26 (2019), 73-89.  doi: 10.5194/npg-26-73-2019.  Google Scholar

[6]

L. De CruzS. SchubertJ. DemaeyerV. Lucarini and S. Vannitsem, Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models, Nonlin. Processes Geophys., 25 (2018), 387-412.  doi: 10.5194/npg-25-387-2018.  Google Scholar

[7]

F. Deutsch, The angle between subspaces of a Hilbert space, in Approximation Theory, Wavelets and Applications, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 454, Kluwer Acad. Publ., Dordrecht, 1995,107-130. doi: 10.1007/978-94-015-8577-4_7.  Google Scholar

[8]

T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, Ph.D thesis, Technische Universität Dresden, 2009. Available from: https://tud.qucosa.de/api/qucosa Google Scholar

[9]

D. Drivaliaris and N. Yannakakis, Subspaces with a common complement in a Banach space, Studia Math., 182 (2007), 141-164.  doi: 10.4064/sm182-2-4.  Google Scholar

[10]

D. Drivaliaris and N. Yannakakis, Subspaces with a common complement in a separable Hilbert space, Integral Equations Operator Theory, 62 (2008), 159-167.  doi: 10.1007/s00020-008-1622-5.  Google Scholar

[11]

G. FroylandT. HülsG. P. Morriss and T. M. Watson, Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39.  doi: 10.1016/j.physd.2012.12.005.  Google Scholar

[12]

G. FroylandS. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.  Google Scholar

[13]

G. FroylandS. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.  Google Scholar

[14]

A. Galántai, Projectors and Projection Methods, Advances in Mathematics (Dordrecht), 6, Kluwer Academic Publishers, Boston, MA, 2004. doi: 10.1007/978-1-4419-9180-5.  Google Scholar

[15]

F. Ginelli, H. Chaté, R. Livi and A. Politi, Covariant Lyapunov vectors, J. Phys. A, 46 (2013), 25pp. doi: 10.1088/1751-8113/46/25/254005.  Google Scholar

[16]

F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.130601.  Google Scholar

[17]

C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, in Contributions of Mexican Mathematicians Abroad in Pure and Applied Mathematics, Contemp. Math., 709, Aportaciones Mat., Amer. Math. Soc., Providence, RI, 2018, 31-52. doi: 10.1090/conm/709/14290.  Google Scholar

[18]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.  Google Scholar

[19]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.  Google Scholar

[20]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, 132, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[21]

M. Lauzon and S. Treil, Common complements of two subspaces of a Hilbert space, J. Funct. Anal., 212 (2004), 500-512.  doi: 10.1016/S0022-1236(03)00253-2.  Google Scholar

[22]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, in Mem. Amer. Math. Soc., 206 (2010). doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[23]

R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in Geometric Dynamics, Lecture Notes in Math., 1007, Springer, Berlin, 1983,522-577. doi: 10.1007/BFb0061433.  Google Scholar

[24]

F. Noethen, Computing Covariant Lyapunov Vectors - A Convergence Analysis of Ginelli's Algorithm, Ph.D thesis, Universität Hamburg, 2019. Available from: https://ediss.sub.uni-hamburg.de/bitstream/ediss/6277/1/Dissertation.pdf. Google Scholar

[25]

F. Noethen, A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors, Phys. D, 396 (2019), 18-34.  doi: 10.1016/j.physd.2019.02.012.  Google Scholar

[26]

F. Noethen, Well-separating common complements of a sequence of subspaces of the same codimension in a Hilbert space are generic, preprint, arXiv: 1906.08514. Google Scholar

[27]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.  Google Scholar

[28]

W. Ott and J. A. Yorke, Prevalence, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 263-290.  doi: 10.1090/S0273-0979-05-01060-8.  Google Scholar

[29]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.  doi: 10.1007/BF02760464.  Google Scholar

[30]

L. Rodman, On global geometric properties of subspaces in Hilbert space, J. Functional Analysis, 45 (1982), 226-235.  doi: 10.1016/0022-1236(82)90020-9.  Google Scholar

[31]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290.  doi: 10.2307/1971392.  Google Scholar

[32]

S. Schubert and V. Lucarini, Covariant Lyapunov vectors of a quasi-geostrophic baroclinic model: Analysis of instabilities and feedbacks, Quart. J. Roy. Meteorol. Soc., 141 (2015), 3040-3055.  doi: 10.1002/qj.2588.  Google Scholar

[33]

K. A. Takeuchi, H.-L. Yang, F. Ginelli, G. Radons and H. Chaté, Hyperbolic decoupling of tangent space and effective dimension of dissipative systems, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.046214.  Google Scholar

[34]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97.  doi: 10.1016/S0294-1449(16)30373-0.  Google Scholar

[35]

A. R. Todd, Covers by linear subspaces, Math. Mag., 63 (1990), 339-342.  doi: 10.1080/0025570X.1990.11977555.  Google Scholar

[36]

S. Vannitsem and V. Lucarini, Statistical and dynamical properties of covariant Lyapunov vectors in a coupled atmosphere-ocean model-Multiscale effects, geometric degeneracy, and error dynamics, J. Phys. A, 49 (2016), 31pp. doi: 10.1088/1751-8113/49/22/224001.  Google Scholar

[37]

C. L. Wolfe and R. M. Samelson, An efficient method for recovering Lyapunov vectors from singular vectors, Tellus A: Dynam. Meteorol. Oceanography, 59 (2007), 355-366.  doi: 10.1111/j.1600-0870.2007.00234.x.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dyn. Syst., 13 (2005), 469-490.  doi: 10.3934/dcds.2005.13.469.  Google Scholar

[3]

C. Blachut and C. González-Tokman, A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems, J. Comput. Dyn., 7 (2020), 369-399.  doi: 10.3934/jcd.2020015.  Google Scholar

[4]

A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2377-2403.  doi: 10.3934/dcds.2016.36.2377.  Google Scholar

[5]

M. CarluF. GinelliV. Lucarini and A. Politi, Lyapunov analysis of multiscale dynamics: The slow bundle of the two-scale Lorenz 96 model, Nonlin. Processes Geophys., 26 (2019), 73-89.  doi: 10.5194/npg-26-73-2019.  Google Scholar

[6]

L. De CruzS. SchubertJ. DemaeyerV. Lucarini and S. Vannitsem, Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models, Nonlin. Processes Geophys., 25 (2018), 387-412.  doi: 10.5194/npg-25-387-2018.  Google Scholar

[7]

F. Deutsch, The angle between subspaces of a Hilbert space, in Approximation Theory, Wavelets and Applications, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 454, Kluwer Acad. Publ., Dordrecht, 1995,107-130. doi: 10.1007/978-94-015-8577-4_7.  Google Scholar

[8]

T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, Ph.D thesis, Technische Universität Dresden, 2009. Available from: https://tud.qucosa.de/api/qucosa Google Scholar

[9]

D. Drivaliaris and N. Yannakakis, Subspaces with a common complement in a Banach space, Studia Math., 182 (2007), 141-164.  doi: 10.4064/sm182-2-4.  Google Scholar

[10]

D. Drivaliaris and N. Yannakakis, Subspaces with a common complement in a separable Hilbert space, Integral Equations Operator Theory, 62 (2008), 159-167.  doi: 10.1007/s00020-008-1622-5.  Google Scholar

[11]

G. FroylandT. HülsG. P. Morriss and T. M. Watson, Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39.  doi: 10.1016/j.physd.2012.12.005.  Google Scholar

[12]

G. FroylandS. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.  Google Scholar

[13]

G. FroylandS. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.  Google Scholar

[14]

A. Galántai, Projectors and Projection Methods, Advances in Mathematics (Dordrecht), 6, Kluwer Academic Publishers, Boston, MA, 2004. doi: 10.1007/978-1-4419-9180-5.  Google Scholar

[15]

F. Ginelli, H. Chaté, R. Livi and A. Politi, Covariant Lyapunov vectors, J. Phys. A, 46 (2013), 25pp. doi: 10.1088/1751-8113/46/25/254005.  Google Scholar

[16]

F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.130601.  Google Scholar

[17]

C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, in Contributions of Mexican Mathematicians Abroad in Pure and Applied Mathematics, Contemp. Math., 709, Aportaciones Mat., Amer. Math. Soc., Providence, RI, 2018, 31-52. doi: 10.1090/conm/709/14290.  Google Scholar

[18]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.  Google Scholar

[19]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.  Google Scholar

[20]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, 132, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[21]

M. Lauzon and S. Treil, Common complements of two subspaces of a Hilbert space, J. Funct. Anal., 212 (2004), 500-512.  doi: 10.1016/S0022-1236(03)00253-2.  Google Scholar

[22]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, in Mem. Amer. Math. Soc., 206 (2010). doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[23]

R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in Geometric Dynamics, Lecture Notes in Math., 1007, Springer, Berlin, 1983,522-577. doi: 10.1007/BFb0061433.  Google Scholar

[24]

F. Noethen, Computing Covariant Lyapunov Vectors - A Convergence Analysis of Ginelli's Algorithm, Ph.D thesis, Universität Hamburg, 2019. Available from: https://ediss.sub.uni-hamburg.de/bitstream/ediss/6277/1/Dissertation.pdf. Google Scholar

[25]

F. Noethen, A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors, Phys. D, 396 (2019), 18-34.  doi: 10.1016/j.physd.2019.02.012.  Google Scholar

[26]

F. Noethen, Well-separating common complements of a sequence of subspaces of the same codimension in a Hilbert space are generic, preprint, arXiv: 1906.08514. Google Scholar

[27]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.  Google Scholar

[28]

W. Ott and J. A. Yorke, Prevalence, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 263-290.  doi: 10.1090/S0273-0979-05-01060-8.  Google Scholar

[29]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.  doi: 10.1007/BF02760464.  Google Scholar

[30]

L. Rodman, On global geometric properties of subspaces in Hilbert space, J. Functional Analysis, 45 (1982), 226-235.  doi: 10.1016/0022-1236(82)90020-9.  Google Scholar

[31]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290.  doi: 10.2307/1971392.  Google Scholar

[32]

S. Schubert and V. Lucarini, Covariant Lyapunov vectors of a quasi-geostrophic baroclinic model: Analysis of instabilities and feedbacks, Quart. J. Roy. Meteorol. Soc., 141 (2015), 3040-3055.  doi: 10.1002/qj.2588.  Google Scholar

[33]

K. A. Takeuchi, H.-L. Yang, F. Ginelli, G. Radons and H. Chaté, Hyperbolic decoupling of tangent space and effective dimension of dissipative systems, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.046214.  Google Scholar

[34]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97.  doi: 10.1016/S0294-1449(16)30373-0.  Google Scholar

[35]

A. R. Todd, Covers by linear subspaces, Math. Mag., 63 (1990), 339-342.  doi: 10.1080/0025570X.1990.11977555.  Google Scholar

[36]

S. Vannitsem and V. Lucarini, Statistical and dynamical properties of covariant Lyapunov vectors in a coupled atmosphere-ocean model-Multiscale effects, geometric degeneracy, and error dynamics, J. Phys. A, 49 (2016), 31pp. doi: 10.1088/1751-8113/49/22/224001.  Google Scholar

[37]

C. L. Wolfe and R. M. Samelson, An efficient method for recovering Lyapunov vectors from singular vectors, Tellus A: Dynam. Meteorol. Oceanography, 59 (2007), 355-366.  doi: 10.1111/j.1600-0870.2007.00234.x.  Google Scholar

Figure 1.  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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