2000/08/28 by Nils Berglund, Barbara Gentz
Environmental Science · Mathematics · Physics and Astronomy · #Chaos control and synchronization #Ecosystem dynamics and resilience #math-ph #math.DS #math.MP #math.PR #msc:34E15 #msc:37H20 #msc:60H10 #msc:93E03 #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.1007/s004400100174
published as Probab. Theory Related Fields 122:341-388 (2002) · 47 pages, 3 figures
arxiv created 2000/08/28 · openalex publication_date 2002/03/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The slow drift (with speed \eps) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity σ, by giving precise estimates on the behaviour of the individual paths. We show that until time √\eps after the bifurcation, the paths are concentrated in a region of size σ/\eps1/4 around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval [√\eps, c√\eps\abslogσ], after which they are likely to stay close to the corresponding deterministic solution. We derive exponentially small upper bounds for the probability of the sets of exceptional paths, with explicit values for the exponents.