2010/04/09 by Stefan Reimann, Reimann, Stefan
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Complex Systems and Time Series Analysis #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #cond-mat.stat-mech #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1004.1619
4 pages, 3 figures
arxiv created 2010/04/09 · openalex publication_date 2010/04/09 · arxiv updated 2010/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The most important characteristic of \em multiplicative noise is that its effects of system's dynamics depends on the recent system's state. Consideration of multiplicative noise on self-referential systems including biological and economical systems therefore is of importance. In this note we study an elementary example. While in a deterministic super critical pitchfork bifurcation with positive bifurcation parameter λ the positive branch √λ is stable, multiplicative white noise λt =λ + σζt on the unique parameter reduces stability in that the system's state tends to 0 almost surely, even for λ>0, while for 'small' noise σ< √(2 λ) the point √(λ-σ2/2) is a meta-stable state. In this case, correspondingly, the system will 'die out', i.e. Xt → 0 within finite time.