Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Pathwise description of dynamic pitchfork bifurcations with additive noise
Download PDF
Download PDF
  • Published: March 2002

Pathwise description of dynamic pitchfork bifurcations with additive noise

  • Nils Berglund1 &
  • Barbara Gentz2 

Probability Theory and Related Fields volume 122, pages 341–388 (2002)Cite this article

  • 298 Accesses

  • 49 Citations

  • 1 Altmetric

  • Metrics details

Abstract.

 The slow drift (with speed ɛ) 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 after the bifurcation, the paths are concentrated in a region of size around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval , 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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA 30332-0430, USA and Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany. Current address: Department of Mathematics, ETH Zürich, ETH Zentrum, 8092 Zürich, Switzerland. e-mail: berglund@math.ethz.ch, , , , , , CH

    Nils Berglund

  2. Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany. e-mail: gentz@wias-berlin.de, , , , , , DE

    Barbara Gentz

Authors
  1. Nils Berglund
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Barbara Gentz
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 7 August 2000 / Revised version: 19 April 2001 / Published online: 20 December 2001

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Berglund, N., Gentz, B. Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab Theory Relat Fields 122, 341–388 (2002). https://doi.org/10.1007/s004400100174

Download citation

  • Issue Date: March 2002

  • DOI: https://doi.org/10.1007/s004400100174

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Additive Noise
  • Pitchfork Bifurcation
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature