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Deterministic and random dynamical systems: theory and numerics

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Modern Methods in Scientific Computing and Applications

Part of the book series: NATO Science Series ((NAII,volume 75))

Abstract

The theory of (random) dynamical systems is a framework for the analysis of large time behaviour of time-evolving systems (driven by noise). These notes contain an elementary introduction to the theory of both dynamical and random dynamical systems. The subject matter is made accessible by means of very simple examples and highlights relationships between the deterministic and the random theories.

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References

  1. L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.

    MATH  Google Scholar 

  2. S. S. Artemiev and T. A. Averina, Numerical Analysis of Systems of Ordinary and Stochastic Differential Equation, VSP Press, Utrecht, 1997.

    Book  Google Scholar 

  3. D. Cheban, P. E. Kloeden and B. Schmalfuß, Pullback attractors in dissipative nonau-tonomous differential equations under discretization, J. Dynam. Differential Equations 13 (2001), 185–213.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), 365–393.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. W. Ford and M. Kac, On the quantum Langevin equation, J. Statist. Phys. 46 (1987), 803–810.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., Providence, RI, 1988.

    Google Scholar 

  7. P. Hartman, Ordinary Differential Equations, Academic Press, New York, 1978.

    Google Scholar 

  8. R. Z. Has’minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980.

    Book  Google Scholar 

  9. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991.

    Book  MATH  Google Scholar 

  10. S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, Academic Press, New York, 1975.

    MATH  Google Scholar 

  11. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Appl. Math. 23, Springer-Verlag, New York, 1992.

    MATH  Google Scholar 

  12. P. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms 14 (1997), 141–152.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. H. Kraichnan, Diffusion by a random velocity field, Phys. Fluids 13 (1970), 22–31.

    Article  MATH  Google Scholar 

  14. A. Majda and P. R. Kramer, Simplified models for turbulent diffusion: theory, numerical modelling and physical phenomena, Phys. Rep. 314 (1999), 237–574.

    Article  MathSciNet  Google Scholar 

  15. X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997.

    MATH  Google Scholar 

  16. J. Mattingly, D. J. Higham, and A. M. Stuart, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, preprint, 2001.

    Google Scholar 

  17. J. Mattingly and A. M. Stuart, Geometric ergodicity of some hypo-elliptic diffusions for particle motions, to appear in Markov Process. Related Fields (2002).

    Google Scholar 

  18. S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, New York, 1992.

    Google Scholar 

  19. B. Øksendal, Stochastic Differential Equations, Springer-Verlag, Berlin, 1998.

    Google Scholar 

  20. J. C. Robinson, Stability of random attractors under perturbation and approximation, submitted to J. Differential Equations (2001).

    Google Scholar 

  21. J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, 1994.

    MATH  Google Scholar 

  22. T. Shardlow and A. M. Stuart, A perturbation theory for ergodic Markov chains with application to numerical approximation, SIAM J. Numer. Anal. 37 (2000), 1120–1137.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Sigurgeirsson and A.M. Stuart, Inertial particles in a random velocity field, submitted to Stochastics Dynamics (2002).

    Google Scholar 

  24. A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.

    MATH  Google Scholar 

  25. D. Talay, Second-order discretization schemes for stochastic differential systems for the computation of the invariant law, Stochastics Stochastics Rep. 29 (1990), 13–36.

    MATH  Google Scholar 

  26. D. Talay, Approximation of upper Lyapunov exponents of bilinear stochastic differential systems, SIAM J. Numer. Anal. 28 (1991), 1141–1164.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Talay, Approximation of the invariant probability measure of stochastic Hamiltonian dissipative systems with non globally Lipschitz coefficients, in: Progress in Stochastic Structural Dynamics (R. Bouc and C. Soize, eds.), Publ. Laboratoire Math. Appl. CNRS 152, 1999.

    Google Scholar 

  28. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag, New York, 1989.

    Google Scholar 

  29. G. E. Uhlenbeck, G. W. Ford, and E. W. Montroll, Lectures in Statistical Mechanics, Lectures in Appl. Math. 1, Amer. Math. Soc., Providence, RI, 1963.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Sussex, Brighton, BN1 9QH, England

    A. R. Humphries

  2. Mathematics Institute, Warwick University, Coventry, CV4 7AL, England

    A. M. Stuart

Authors
  1. A. R. Humphries
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  2. A. M. Stuart
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Editor information

Editors and Affiliations

  1. Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada

    Anne Bourlioux  & Gert Sabidussi  & 

  2. Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada

    Martin J. Gander

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© 2002 Springer Science+Business Media Dordrecht

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Humphries, A.R., Stuart, A.M. (2002). Deterministic and random dynamical systems: theory and numerics. In: Bourlioux, A., Gander, M.J., Sabidussi, G. (eds) Modern Methods in Scientific Computing and Applications. NATO Science Series, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0510-4_6

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  • DOI: https://doi.org/10.1007/978-94-010-0510-4_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0782-8

  • Online ISBN: 978-94-010-0510-4

  • eBook Packages: Springer Book Archive

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