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Bounded Noises in Physics, Biology, and Engineering pp 133–149Cite as

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Bifurcations of Random Differential Equations with Bounded Noise

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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.

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Authors and Affiliations

  1. KdV Institute for Mathematics, University of Amsterdam, Science park 904, 1098 XH, Amsterdam, The Netherlands

    Ale Jan Homburg & Masoumeh Gharaei

  2. Department of Mathematics, VU University Amsterdam, De Boelelaan 1081, HV Amsterdam, The Netherlands

    Ale Jan Homburg

  3. Department of Mathematics, Ohio University, Morton Hall, Athens, OH, 45701, USA

    Todd R. Young

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  1. Ale Jan Homburg
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  2. Todd R. Young
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  3. Masoumeh Gharaei
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Correspondence to Todd R. Young .

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Editors and Affiliations

  1. Department of Experimental Oncology, European Institute of Oncology, Milan, Italy

    Alberto d'Onofrio

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Homburg, A.J., Young, T.R., Gharaei, M. (2013). Bifurcations of Random Differential Equations with Bounded Noise. In: d'Onofrio, A. (eds) Bounded Noises in Physics, Biology, and Engineering. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7385-5_9

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