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Small fluctuations at the unstable steady state

University of British Columbia

The effects of small random fluctuations on deterministic systems are studied. The deterministic systems of interest have multiple steady states. As parameters vary, two or three of the steady states coalesce. This work is concerned with the long time behavior of the system, when the system starts near an unstable steady state. The deterministic separatrix is surrounded by a tube that contains up to two stable steady states. The quantity of basic interest is the probability of first exit from the tube through a specified boundary, conditioned on initial position. In the diffusion approximation this probability satisfies a backward diffusion equation. Formal asymptotic solutions of the backward equation are constructed. The solutions are obtained by a generalized "ray method" and are given in terms of various incomplete special functions. As an example, the effects of fluctuations on a substrate inhibited reaction in an open vessel are analyzed. The theory is compared with exact solutions, for a one dimensional model; and Monte Carlo experiments, for a two dimensional model.

Text

2010-03-06

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http://hdl.handle.net/2429/21636

Mathematics

University of British Columbia

Graduate