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Computer-Assisted Proofs of Hopf Bubbles and Degenerate Hopf Bifurcations
Kevin Church1, Elena Queirolo2
1Université de Montréal, Montreal, Canada.
Journal of Dynamics and Differential Equations
|November 18, 2024
Summary
This study introduces a computer-assisted method to prove Hopf bubbles and degenerate Hopf bifurcations in differential equations. The approach rigorously analyzes nonlocal orbit structures in complex biological and climate models.
Area of Science:
- Dynamical Systems and Mathematical Biology
- Computational Mathematics
Background:
- Hopf bifurcations are critical for understanding system dynamics.
- Nonlocal orbit structures like Hopf bubbles are complex phenomena.
- Proving their existence in differential equations is challenging.
Purpose of the Study:
- To develop and present a novel computer-assisted approach.
- To rigorously prove the existence of Hopf bubbles and degenerate Hopf bifurcations.
- To apply this method to significant mathematical models.
Main Methods:
- Utilizing a computer-assisted proof strategy.
- Applying the method to ordinary and delay differential equations.
- Investigating nonlocal orbit structures.
Main Results:
- Successfully proved the existence of Hopf bubbles.
- Demonstrated the presence of degenerate Hopf bifurcations.
- Rigorously analyzed these structures in the FitzHugh-Nagumo equation, Lorenz-84 model, and a time-delay SI model.
Conclusions:
- The computer-assisted approach is effective for proving complex bifurcations.
- This method provides rigorous analysis of nonlocal orbit structures.
- The findings advance the understanding of dynamics in the studied models.
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