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The Lyapunov Characteristic Exponents and Applications to the Dimension of the Invariant Manifolds and Chaotic Attractors

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Stability of the Solar System and Its Minor Natural and Artificial Bodies

Part of the book series: NATO ASI Series ((ASIC,volume 154))

Abstract

After a presentation of Lyapunov characteristic exponents (LCE) we recall their basic properties and numerical methods of computation. We review some numerical computations which are concerned with LCEs mainly those concerning the dimensions of invariant manifolds and chaotic attractors.

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References

  1. Chaotic behaviour of deterministic systems, Cours des Houches XXXVI, North Holland, 1981.

    Google Scholar 

  2. Arnold V.I., Mathematical methods of Classical Mechanics, Ed de Moscou, 1976.

    Google Scholar 

  3. Benettin G. and Galgani L., ‘Lyapunov characteristic exponents and Stochasticity, intrinsic stochasticity in Plasma’ edited by Laval G. and Gresillon D. Distributor: les Editions de Physique Courtaboeuf Orsay, France, p. 94–114, 1979

    Google Scholar 

  4. Benettin G., Froeschlé C., Scheidecker J.P., ‘Kolmogorov Entropy of Dynamical systems with increasing number of degrees of freedom’, Phys. Rev. A 19, p. 2454–2460, 1979.

    Article  MathSciNet  ADS  Google Scholar 

  5. Benettin G., Galgani L., Giorgilli A., Strelcyn J.M., ‘Lyapunov characteristic exponents for smooth Dynamical systems; a method for computing all of them. Part 1: Theory, p. 9–20; Part 2: Numerical applications, Meccanica March, p, 21–30, 1980.

    Google Scholar 

  6. Cesari L., Asymptotic Behaviour and Stability Problems in Ordinary differential equations, Springer Verlag Berlin, 1959.

    Google Scholar 

  7. Chirikov B.V., ‘An universal instability of many dimensional oscillator systems, Phys. Rep. 52, p. 263–379, 1979.

    Article  MathSciNet  ADS  Google Scholar 

  8. Farmer J.D., Otte and Yorke J., ‘The dimension of chaotic attractors’ Physica 7D, p. 153–180, 1983.

    ADS  Google Scholar 

  9. Ford J., ‘The Statistical Mechanics of Classical Analytic Dynamics’ Fundamental problems in Statistical Mechanics, ed. E.G.D. Cohen, Vol III (North Holland, Amterdam), p. 215–255, 1975.

    Google Scholar 

  10. Froeschlé C., ‘Numerical Study of Dynamical systems with three degrees of Freedom, I Graphical Displays of four-dimensional section’, Astron. Astrophys. 4, p. 115–128, 1970,

    ADS  Google Scholar 

  11. Froeschlé C., ‘Numerical Study of Dynamical systems with three degrees of Freedom, II Numerical Displays of four-dimensional sections’, Astron. Astrophys. 5 p. 177–183, 1970.

    ADS  Google Scholar 

  12. Froeschlé C., ‘A numerical Study of the Stochasticité of Dynamical Systems with two degrees of Freedom’, Astron. Astrophys. 9, p. 15–23, 1970.

    ADS  MATH  Google Scholar 

  13. Froeschlé C. and Scheidecker J.P., ‘On the disappearance of isolating integrals in systems with more than two degrees of Freedom’, Astrophys. and Space Sc. 25, p. 373–386, 1973.

    Article  ADS  MATH  Google Scholar 

  14. Froeschlé C. and Scheidecker J.P. ‘Numerical study of the stochasticity of Dynamical System with more than two degrees of Freedom’, J. Comp. Phys. Vol. 11, n° 3, p. 423–439, 1973.

    Article  ADS  Google Scholar 

  15. Gonczi R., Froeschlé C., ‘The Lyapunov characteristic exponents as indicators of stochasticity in the restricted Three-body Problem’, Cel. Mech. 25, p. 271–280, 1981.

    Article  ADS  MATH  Google Scholar 

  16. Guckenheimer J., Moser J., New House S., Dynamical Systems, CIME lectures Birhauser, 1978.

    Google Scholar 

  17. Hénon M. and Heiles C., ‘The applicatibility of the third integral of motion, some numerical experiments’, Astron. Journal 69, p. 73–79, 1964.

    Article  ADS  Google Scholar 

  18. Kaplan J. and York J., ‘Functional differential equations and the approximation of fixed points’, Proceedings, Bonn, July, Lectures Notes in Math. 730, H.O. Dietten and H.O. Walter, eds., Springer Berlin, 1978.

    Google Scholar 

  19. Kolmogorov A.N., ‘A new invariant for transitive dynamical systems’ DOK1, Akad. Nank SSSR 119, p. 861–864, 1958.

    MathSciNet  MATH  Google Scholar 

  20. Oseledec V.I., ‘A multiplicative ergodic theorem. The Lyapunov characteristic numbers of Dynamical system’ (in Russian) Trudy Mosk. Mat. Obsc. 19, p.179–210, 1968. English translation. Mosc. Math. Soc. 19, p. 197–231, 1968.

    MathSciNet  Google Scholar 

  21. Russel D.A., Hanson J.D., and Ott E., ‘The dimension of strange attractors’, Rev. Lett. 45, p. 1175–1178, 1980.

    Article  ADS  Google Scholar 

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© 1985 D. Reidel Publishing Company

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Froeschlé, C. (1985). The Lyapunov Characteristic Exponents and Applications to the Dimension of the Invariant Manifolds and Chaotic Attractors. In: Szebehely, V.G. (eds) Stability of the Solar System and Its Minor Natural and Artificial Bodies. NATO ASI Series, vol 154. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5398-7_21

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  • DOI: https://doi.org/10.1007/978-94-009-5398-7_21

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8883-1

  • Online ISBN: 978-94-009-5398-7

  • eBook Packages: Springer Book Archive

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