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Magnetically Induced Rotating Rayleigh-Taylor Instability
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On a stable torus in a 3D system with a saddle-focus.

Andrey L Shilnikov1, Leonid P Shilnikov2

  • 1Neuroscience Institute and Department of Mathematics and Statistics, Georgia State University, 100 Piedmont Ave., Atlanta, Georgia 30303, USA.

Chaos (Woodbury, N.Y.)
|July 15, 2026
PubMed
Summary

This study introduces a model for stable torus formation near saddle-focus points, crucial for understanding elliptic bursting in mathematical neuroscience models. The research also explores related dynamical regimes and homoclinic bifurcations.

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Area of Science:

  • Mathematical Neuroscience
  • Dynamical Systems Theory
  • Computational Neuroscience

Background:

  • Elliptic bursting is a common firing pattern in neurons.
  • This pattern is often generated by slow-fast dynamical systems.
  • Saddle-focus equilibria are key to understanding complex neuronal dynamics.

Purpose of the Study:

  • Propose a conceptual model for the onset of a stable torus near a saddle-focus equilibrium.
  • Investigate bifurcation scenarios in slow-fast systems relevant to neuronal bursting.
  • Explore other dynamical regimes and homoclinic bifurcations.

Main Methods:

  • Development of a conceptual model.
  • Analysis of bifurcation scenarios.
  • Examination of slow-fast systems and neuronal models.

Main Results:

  • A model for stable torus onset near saddle-focus points was proposed.
  • The model captures elliptic bursting typical of neuronal dynamics.
  • Related dynamical regimes and homoclinic bifurcations were analyzed.

Conclusions:

  • The conceptual model provides insights into neuronal bursting mechanisms.
  • The findings are applicable to various mathematical neuroscience models.
  • The study contributes to understanding complex dynamics near saddle-focus equilibria.