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)


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.


Asymptotic Expansion Lyapunov Exponent Maximal Lyapunov Exponent Real Noise Random Dynamical System 
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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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