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Additive noise destroys a pitchfork bifurcation. (English) Zbl 0907.34042
Summary: In the deterministic pitchfork bifurcation the dynamical behavior of a system changes as the parameter crosses the bifurcation point. The stable fixed point loses its stability. Two new stable fixed points appear. The respective domains of attraction of those two fixed split the state space into two macroscopically distinct regions. It is shown that this bifurcation of the dynamical behavior disappears as soon as additive white noise of arbitrarily small intensity is incorporated into the model. The dynamical behavior of the disturbed system remains the same for all parameter values. In particular, the system has a (random) global attractor, and this attractor is a one-point set for all parameter values. For any parameter value all solutions converge to each other almost surely (uniformly in bounded sets). No splitting of the state space into distinct regions occurs, not even into random ones. This holds regardless of the intensity of the disturbance.

MSC:
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34C23 Bifurcation theory for ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
60D05 Geometric probability and stochastic geometry
93E03 Stochastic systems in control theory (general)
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