A Markov Chain Approximation of the Stochastic Dynamical System
Part of the Springer Series in Synergetics book series (SSSYN)
Dealing with a stochastic dynamical system with state vector X(t) satisfying a nonlinear Langevin equation of the type
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.
$$dx = f(x)dt + \sigma (x)dW(t),$$
KeywordsAtmospheric Circulation Preferent State Markov Chain Model Spectral Model Occupation Rate
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© Springer-Verlag Berlin Heidelberg 1999