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Stability of knots in excitable media
Paul M Sutcliffe1, Arthur T Winfree
1Institute of Mathematics, University of Kent, Canterbury CT2 7NF, United Kingdom.
Summary
Numerical simulations reveal that knotted vortices in the FitzHugh-Nagumo model maintain their topology over time. However, their shape evolves, leading to metastable configurations dependent on initial conditions.
Area of Science:
- Complex systems
- Nonlinear dynamics
- Computational physics
Background:
- Vortices are fundamental structures in fluid dynamics and reaction-diffusion systems.
- Understanding vortex dynamics, including topological configurations like knots and links, is crucial for various scientific fields.
- The FitzHugh-Nagumo model provides a simplified yet powerful framework for studying complex spatiotemporal patterns.
Purpose of the Study:
- To investigate the long-term evolution of knotted and linked vortices within the FitzHugh-Nagumo model.
- To analyze how vortex topology and shape change over time under different conditions.
- To identify the factors influencing the asymptotic states of these complex vortex structures.
Main Methods:
- Extensive numerical simulations were performed to model vortex evolution.
- The study tracked vortex translation, precession, and shape changes over medium and long time scales.
- Analysis focused on topological preservation and the emergence of asymptotic configurations.
Main Results:
- Knotted vortices translate and precess with minimal shape change on medium time scales.
- On longer time scales, vortex arcs expand and contract, causing significant length variations.
- The topological structure of knots remains invariant throughout the evolution.
- An asymptotic state is reached, which is contingent on initial conditions, indicating multiple metastable configurations.
Conclusions:
- Vortex topology is preserved in the FitzHugh-Nagumo model, but shape evolves towards metastable states.
- The asymptotic configurations are not unique, even for a given topology, highlighting the complexity of vortex dynamics.
- Higher-frequency wavefronts may drive the observed evolutionary mechanisms in knotted vortices.
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