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An elementary model of torus canards
G Nicholas Benes1, Anna M Barry, Tasso J Kaper
1Department of Mathematics and Statistics, Center for BioDynamics, Boston University, Boston, Massachusetts 02215, USA.
Chaos (Woodbury, N.Y.)
|July 5, 2011
Summary
Torus canards, complex trajectories in fast-slow systems, exhibit unique dynamics. This study reveals how phase dependence in a simple model generates rich torus canard and mixed-mode behaviors, offering insights into neuroscience models.
Area of Science:
- Dynamical Systems Theory
- Nonlinear Dynamics
- Mathematical Neuroscience
Background:
- Canard orbits are classical phenomena in planar fast-slow systems.
- Torus canards are higher-dimensional generalizations observed in systems with saddle-node bifurcations of limit cycles.
- These trajectories exhibit complex behavior, spending extended periods near repelling limit cycles.
Purpose of the Study:
- To investigate torus canard dynamics in an elementary third-order system.
- To analyze the influence of broken rotational symmetry and phase dependence on torus canards.
- To provide insights into torus canards observed in complex neuroscience models.
Main Methods:
- Analysis of a third-order ordinary differential equation system.
- Study of a rotated van der Pol type system with a phase-dependent term.
- Examination of dynamics in fast and slow rotation regimes.
Main Results:
- In fast rotation, torus canards resemble planar counterparts.
- Slow rotation with phase dependence leads to rich torus canard dynamics and mixed-mode behaviors.
- The elementary model successfully replicates key aspects of torus canard phenomena.
Conclusions:
- Phase dependence is crucial for generating complex torus canard dynamics.
- The studied third-order system serves as a valuable model for understanding higher-dimensional torus canards.
- Findings offer a simplified yet insightful perspective on torus canards in neuroscience applications.
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