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A Markov Chain Approximation of the Stochastic Dynamical System

  • Johan Grasman
  • Onno A. van Herwaarden
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

Dealing with a stochastic dynamical system with state vector X(t) satisfying a nonlinear Langevin equation of the type
$$dx = f(x)dt + \sigma (x)dW(t),$$
we may only be interested in qualitative changes and their statistics. Typical qualitative changes are the switch from one domain of attraction to a neighbouring one or in the case of population dynamics the extinction or introduction of a species. For that purpose we introduce the Markov chain approximation. For a precise mathematical description of the connection between such a Markov chain and the stochastic dynamical system we refer to Chap. 6 of Freidlin and Wentzell (1984). Here we work out two examples. One modelling the preferent states of the atmospheric circulation and one dealing with the occupation rate of a patch of land by some biological species.

Keywords

Atmospheric Circulation Preferent State Markov Chain Model Spectral Model Occupation Rate 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Johan Grasman
    • 1
  • Onno A. van Herwaarden
    • 1
  1. 1.Department of MathematicsAgricultural UniversityWageningenThe Netherlands

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