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Bifurcations of Random Differential Equations with Bounded Noise

  • Ale Jan Homburg
  • Todd R. Young
  • Masoumeh Gharaei
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

We review recent results from the theory of random differential equations with bounded noise. Assuming the noise to be “sufficiently robust in its effects” we discuss the feature that any stationary measure of the system is supported on a “Minimal Forward Invariant” (MFI) set. We review basic properties of the MFI sets, including their relationship to attractors in systems where the noise is small. In the main part of the paper we discuss how MFI sets can undergo discontinuous changes that we have called hard bifurcations. We characterize such bifurcations for systems in one and two dimensions and we give an example of the effects of bounded noise in the context of a Hopf–Andronov bifurcation.

Keywords

Bounded noises Random differential equations Stationary measures Stochastic bifurcations Hopf–Andronov bifurcation Hard bifurcations 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ale Jan Homburg
    • 1
    • 2
  • Todd R. Young
    • 3
  • Masoumeh Gharaei
    • 1
  1. 1.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Department of MathematicsVU University AmsterdamHV AmsterdamThe Netherlands
  3. 3.Department of MathematicsOhio UniversityAthensUSA

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