ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


Cite this article as:

Kuptsov P. V. Computation of Lyapunov exponents for spatially extended systems: advantages and limitations of various numerical methods. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 5, pp. 91-110. DOI: https://doi.org/10.18500/0869-6632-2010-18-5-91-110

Language: 
Russian
Heading: 

Computation of Lyapunov exponents for spatially extended systems: advantages and limitations of various numerical methods

Autors: 
Kuptsov Pavel Vladimirovich, Yuri Gagarin State Technical University of Saratov
Тип статьи для РИНЦ: 
RAR научная статья
Abstract: 

Problems emerging in computations of Lyapunov exponents for spatially extended systems are considered. We concentrate on the incorrect orthogonalization of high sized ill conditioned matrices appearing in course of the computation, and on large errors emerging under certain conditions if the finite difference numerical method is applied to solve equations. The practical guidelines helping to avoid the mentioned problems are represented.

Key words: 
DOI: 
10.18500/0869-6632-2010-18-5-91-110
References: 

1. Оселедец В.И. Мультипликативная эргодическая теорема. характеристические показатели Ляпунова динамических систем // Труды Моск. матем. об-ва. 1968. T. 19. С. 197. 2. Eckmann J.P., Ruelle D. Ergodic theory of chaos and strange attractors// Rev. Mod. Phys. 1985. Jul. Vol. 57, No 3. P 617. 3. Schaumloffel K.-U.  ? Multiplicative ergodic theorems in infinite dimensions // Lyapu-nov Exponents. Springer Berlin / Heidelberg, 1991. T. 1486/1991. Lecture Notes in Mathematics. C. 187. 4. Robinson J.C. Finite dimensional behavior in dissipative partial differential equations // Chaos. 1995. Vol. 5. P. 330. 5. Yang H.-L., Takeuchi K.A., Ginelli F., Chate H., Radons G.  ? Hyperbolicity and the effective dimension of spatially-extended dissipative systems // Phys. Rev. Lett. 2009. Vol. 102. P. 074102. 6. Kuptsov P.V., Parlitz U. Strict and fussy modes splitting in the tangent space of the Ginzburg–Landau equation // Phys. Rev. E. 2010. Vol. 81. P. 036214. 7. Benettin G., Galgani L., Giorgilli A., Strelcyn J.M. Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems: A method for computing all of them. Part I: Theory. Part II: Numerical application // Meccanica. 1980. Vol. 15. P. 9. 8. Parker T.S., Chua L.O. Practical numerical algorithms for chaotic systems. Springer-Verlag, 1989. P. 348. 9. Golub G.H., van Loan C.F. Matrix computations. Third Edition. The Johns Hopkins University Press, Baltimore, MD. 1996. P. 694. 10. Geist K., Parlitz U., Lauterborn W. Comparision of different methods for computing Lyapunov exponents // Prog. Theor. Phys. 1990. Vol. 83, No 5. P. 875. 11. Numerical recipes in C / W.H. Press, S.A. Teukolsky, W.T. Vettering, B.P. Flannery. Cambridge University Press, 1992. P. 994. 12. Aranson I.S., Kramer L. The world of the complex Ginzburg–Landau equation // Rev. Mod. Phys. 2002. Vol. 74. P. 99. 13. Калиткин Н.Н. Численные методы: Учебное пособие. М.: Наука, 1978. C. 512.

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