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Sektion 7
Dienstag, 19.09.2000, 14.30–14.50 Uhr, POT 112

Dynamic pitchfork bifurcations with additive noise

Barbara Gentz, Weierstrass-Institut Berlin

Consider the stochastic differential equation

dxs = f (xs, c)ds + s dWs,
where f is such that the dynamical system given by
dxs = f(xs,c) ds
undergoes a pitchfork bifurcation. The standard example is f(x, c) = cx - x3,  x  (- R. Crauel and Flandoli (1998) showed that arbitrarily small noise s > 0 in (1) destroys the bifurcation undergone by the deterministic system (2).

Here, we are interested in dynamic bifurcations. Instead of assuming that c is constant, we allow c to vary slowly in time, c = es for small e > 0. We introduce the slow time t = es and study the corresponding systems

       1             s
dxt =  -f(xt,t)dt +  V~ --dWt
       e              e
and
 dx
e-dt = f(x, t).
It is known that the behaviour of the dynamical system (4) differs from the behaviour of (2) by showing a bifurcation delay: As the bifurcation parameter passes through the bifurcation point, the solutions of (4) remain close to the unstable equilibrium at zero for some time, before they track one of the stable equilibrium branches of (2), while the solutions of (2) immediately approach a stable equilibrium.

For small, but not exponentially small noise, we show that with high probability, the solutions of (3) follow the solutions of the corresponding deterministic system (4) up to time O( V~ --
  e). Already after time O( V~ __________e| log s|), they track one of the stable equilibrium branches of the static system (2). This shows that the bifurcation delay is destroyed by small additive noise as soon as the noise is not exponentially small.

This is joint work with Nils Berglund, WIAS Berlin.