A stochastic Hopf bifurcation

Abstract

Let {xt:t≧0} be the solution of a stochastic differential equation (SDE) in ℝd which fixes 0, and let λ 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 λ controls the stability/instability of 0 and the transience/recurrence of {xt:t≧0} on ℝd\{0}. In particular if the coefficients in the SDE depend on some parameter z which is varied in such a way that the corresponding Lyapunov exponent λz 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 μz on ℝd\{0}. In this paper we investigate the limiting behavior of μz as λz converges to 0 from above. The main result is that the rescaled measures (1/λz)μz converge (in an appropriate weak sense) to a non-trivial σ-finite measure on ℝd\{0}. © 1994 Springer-Verlag.