An exponentially-fitted method for singularly-perturbed ordinary differential equations with turning points and parabolic problems

2005 ◽  
Vol 165 (3) ◽  
pp. 549-564 ◽  
Author(s):  
J.I. Ramos
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Turning Points ◽  
Singularly Perturbed ◽  
Parabolic Problems ◽  
Exponentially Fitted ◽  
Exponentially Fitted Method

NUMERICAL COMPUTATION OF CANARDS

2000 ◽  
Vol 10 (12) ◽  
pp. 2669-2687 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Electrical Potential ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Chemical Systems ◽  
Perturbed System ◽  
Physical And Chemical

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

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