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Small Noise Expansion of Moment Lyapunov Exponents for Two-Dimensional Systems

  • M. M. Doyle
  • N. Sri Namachchivaya
  • L. Arnold
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 47)

Abstract

Sample or almost-sure stability of a stationary solution of a random dynamical system is of importance in the context of dynamical systems theory since it guarantees all samples except for a set of measure zero tend to the stationary solution as time goes to infinity. The almost-sure stability or instability of a dynamical system is indicated by the sign of the maximal Lyapunov exponent. However, from the applications viewpoint, one may not be satisfied with such guarantees since a sample stable process may still exceed some threshold values or may possess a slow rate of decay. Although sample solutions may be stable with probability one, the mean square response of the system for the same parameter values may grow exponentially. It is well known that there are parameter values at which the top Lyapunov exponent λ is negative, indicating that the system is sample stable, while the p th moments grow exponentially for large p indicating the p th mean response is unstable.

Keywords

Asymptotic Expansion Lyapunov Exponent Maximal Lyapunov Exponent Real Noise Random Dynamical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Arnold, L. (In preparation) Randon Dynamical Systems.Google Scholar
  2. 2.
    Arnold, L. (1984) A formula connecting sample and moment stability of linear stochastic systems, SIAM Journal of Applied Mathematics, 44(4), pp. 793–802.zbMATHCrossRefGoogle Scholar
  3. 3.
    Arnold, L. and Kliemann, W. (1987) Large deviations of linear stochastic differential equations, vol. 96 of Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, pp. 117–151.Google Scholar
  4. 4.
    Arnold, L., Kliemann, W. and Oeljeklaus, E. (1986) Lyapunov exponents of linear stochastic systems, vol. 1186 of Lecture Notes in Mathematics, Springer-Verlag, New York, pp. 85–125.Google Scholar
  5. 5.
    Arnold, L., Oeljeklaus, E. and Pardoux, E. (1986) Almost sure and moment stability for linear Itô equations, vol. 1186 of Lecture Notes in Mathematics, Springer-Verlag, New York, pp. 129–159.Google Scholar
  6. 6.
    Arnold, L. and Wihstutz, V. (1986) “Lyapunov exponents”, vol. 1186 of Lecture Notes in Mathematics, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  7. 7.
    Baxendale, P. H. (1987) Moment stability and large deviations for linear stochastic differential equations, in Taniguchi Symposium PMPP.Google Scholar
  8. 8.
    Baxendale, P. H. (1990) Invariant measures for nonlinear stochastic differential equations, vol. 1486 of Lecture Notes in Mathematics, Springer-Verlag, New York, pp. 123–140.Google Scholar
  9. 9.
    Doyle, M. M. and N. Sri Namachchivaya (1994) Almost-sure asymptotic stability of a general four dimensioned dynamical system driven by real noise, Journal of Statistical Physics, 75(3/4), pp. 525–552.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Doyle, M. M., N. Sri Namachchivaya and L. Arnold (Submitted for publication) Small noise expansion of moment Lyapunov exponents for general two dimensional systems.Google Scholar
  11. 11.
    Khas’minskii, R. Z. (1980) Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn. (Translation of the Russian edition, Nauka, Moscow, 1969)Google Scholar
  12. 12.
    Kozin, F. and S. Sugimoto (1977) Relations between sample and moment stability for linear stochastic differential equations, in J. David Maison (ed.), Proceedings of Conference on Stochastic Differential Equations and Applications, Academic Press, pp. 145–162.Google Scholar
  13. 13.
    Molcanov, S. A. (1978) The structure of eigenfunctions of one-dimensional unordered structures, Math. USSR Izvestija, 12(1), pp. 69–101.CrossRefGoogle Scholar
  14. 14.
    Pardoux, E. and V. Wihstutz (1988) Lyapunov exponent and rotation number of two-dimensional linear stochastic aystems with amall diffusion, SIAM Journal of Applied Mathematics, 48(2), pp. 442–457.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Pinsky, M. A. (1992) Lyapunov exponent and rotation number of the linear harmonic oscillator, in M. A. Pinsky and V. Wihstutz (eds.), Diffusion Processes and Related Problems in Analysis, Volume II (Stochastic Flows) Birkhâuser, Boston”, pp. 257–267.CrossRefGoogle Scholar
  16. 16.
    Sri Namachchivaya, N. and Ariaratnam, S. T. (1987) Stochastically perturbed Hopf bifurcation International Journal of Non-Linear Mechanics 22(5), pp. 363–372 (see also Stochastically perturbed Hopf bifurcation, in F. M. A. Salam and M. L. Levi (eds.), Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits, SIAM, pp. 39–52, 1988).zbMATHCrossRefGoogle Scholar
  17. 17.
    Sri Namachchivaya, N. and H. J. Van Roessel, Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by reed noise, Journal of Statistical Physics, 71(3/4), pp. 549–567.Google Scholar
  18. 18.
    Stroock, D. W. (1986) On the rate at which a homogeneous diffusion approaches a limit, an application of the large deviation theory of certain stochastic integrals, Annals of Probability, 14, pp. 840–859.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • M. M. Doyle
    • 1
  • N. Sri Namachchivaya
    • 1
  • L. Arnold
    • 2
  1. 1.Nonlinear Systems Group, Department of Aeronautical and Astronautical EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaGermany
  2. 2.Institute for Dynamical Systems, University of BremenBremen 33Germany

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