A stochastic Hopf bifurcation

Summary Let {x_t:t0} be the solution of a stochastic differential equation (SDE) in ^d which fixes 0, and let denote the Lyapunov exponent for the linear SDE obtained by linearizing the original SDE at 0. It is known that, under appropriate conditions, the sign of controls the stability/instability of 0 and the transience/recurrence of {x_t:t0} on ^d{0}. In particular if the coefficients in the SDE depend on some parameterz which is varied in such a way that the corresponding Lyapunov exponent^z changes sign from negative to positive the (almost-surely) stable fixed point at 0 is replaced by an (almost-surely) unstable fixed point at 0 together with an attracting invariant probability measure^z on ^d{0}. In this paper we investigate the limiting behavior of^z as^z converges to 0 from above. The main result is that the rescaled measures (1/^z)^z converge (in an appropriate weak sense) to a non-trivial -finite measure on ^d{0}.Research supported in part by Office of Naval Research contract N00014-91-J-1526