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Sealable Femtoliter Chamber Arrays for Cell-free Biology
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THE HOPF BIFURCATION WITH BOUNDED NOISE.

Ryan T Botts1, Ale Jan Homburg2, Todd R Young3

  • 1Department of Mathematical, Information & Computer Sciences, Point Loma Nazarene University, 3900 Lomaland Drive, San Diego, CA 92106, USA.

Discrete and Continuous Dynamical Systems. Series A
|April 22, 2014
PubMed
Summary

We investigated Hopf-Andronov bifurcations in random differential equations with bounded noise. Bounded noise causes a discontinuous change in the minimal forward invariant set during these bifurcations.

Keywords:
Random dynamical systemminimal forward invariant setrandom differential equationstationary measure

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Area of Science:

  • Dynamical systems
  • Stochastic analysis
  • Bifurcation theory

Background:

  • Hopf bifurcations are critical in understanding the qualitative changes in dynamical systems.
  • Random differential equations (RDEs) offer a more realistic model for systems influenced by noise.
  • The effect of bounded noise on bifurcations is not fully understood.

Purpose of the Study:

  • To analyze Hopf-Andronov bifurcations in RDEs with bounded noise.
  • To characterize the impact of bounded noise on the dynamics of these systems.
  • To identify changes in invariant sets during noisy bifurcations.

Main Methods:

  • Studying a class of random differential equations with specific bounded noise characteristics.
  • Applying bifurcation theory to analyze the qualitative behavior of the system.
  • Investigating the properties of the Minimal Forward Invariant set under noise perturbation.

Main Results:

  • Hopf-Andronov bifurcations were observed in the studied RDEs.
  • The presence of bounded noise leads to a discontinuous change in the Minimal Forward Invariant set.
  • This discontinuity signifies a distinct type of bifurcation under noisy conditions.

Conclusions:

  • Bounded noise fundamentally alters the nature of Hopf-Andronov bifurcations.
  • The Minimal Forward Invariant set serves as a key indicator of these noisy bifurcations.
  • Findings contribute to the understanding of stochastic dynamical systems and bifurcation phenomena.