Small fluctuations at the unstable steady state

The e f fec t s of small random f l u c t u a t i o n s on d e t e r m i n i s t i c systems are s tud ied . The d e t e r m i n i s t i c systems of i n t e r e s t have m u l t i p l e steady s ta te s . As parameters v a r y , two or three of the steady states coalesce . This work i s concerned with the long time behavior of the system, when the system s t a r t s near an unstable steady s ta te . The d e t e r m i n i s t i c s eparatr ix i s sur rounded by a tube that contains up to two stable steady s ta te s . The quant i ty of bas ic i n t e r e s t i s the p r o b a b i l i t y of f i r s t ex i t from the tube through a s p e c i f i e d boundary, condit ioned on i n i t i a l p o s i t i o n . In the d i f f u s i o n approximation t h i s p r o b a b i l i t y s a t i s f i e s a backward d i f f u s i o n equation. Formal asymptotic so lu t ions of the backward equation are constructed . The so lut ions are obtained by a general ized "ray method" and are given i n terms of var ious incomplete s p e c i a l func t ions . As an example, the e f fec t s of f l u c t u a t i o n s on a substrate i n h i b i t e d r e a c t i o n i n an open ves se l are analyzed. The theory i s compared with exact s o l u t i o n s , for a one dimensional model; and Monte Carlo experiments, for a two dimensional model.