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.

[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.

[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.

[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.

[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.

[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.

[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.

[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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[35]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[35]

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

[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.

[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.

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
[1]

Lucas Backes. On the periodic approximation of Lyapunov exponents for semi-invertible cocycles. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6353-6368. doi: 10.3934/dcds.2017275

[2]

Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835

[3]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237

[4]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247

[5]

Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377

[6]

Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems and Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019

[7]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109

[8]

Mateusz Krukowski. Arzelà-Ascoli's theorem in uniform spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 283-294. doi: 10.3934/dcdsb.2018020

[9]

Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325

[10]

Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247

[11]

David Brander. Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane. Electronic Research Announcements, 2008, 15: 8-16. doi: 10.3934/era.2008.15.8

[12]

Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227

[13]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197

[14]

Aviv Gibali, Dang Thi Mai, Nguyen The Vinh. A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. Journal of Industrial and Management Optimization, 2019, 15 (2) : 963-984. doi: 10.3934/jimo.2018080

[15]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103

[16]

Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067

[17]

Zhiming Li, Yujun Zhu. Entropies of commuting transformations on Hilbert spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5795-5814. doi: 10.3934/dcds.2020246

[18]

Shrey Sanadhya. A shrinking target theorem for ergodic transformations of the unit interval. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4003-4011. doi: 10.3934/dcds.2022042

[19]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367

[20]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics and Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011

 Impact Factor: 

Metrics

  • PDF downloads (79)
  • HTML views (344)
  • Cited by (0)

Other articles
by authors

[Back to Top]