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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
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.
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.
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