Construction of the Fitzhugh-Nagumo Pulse Using Differential Forms

Systems of singularly perturbed ordinary differential equations can often be solved approximately by singular solutions. These singular solutions are pieced together from solutions to simpler sets of equations obtained as limits from the original equations. There is a large body of literature on the question of when the existence of a singular solution implies the existence of an actual solution to the original equations. Techniques that have been used include fixed point arguments (Conley [1], Carpenter [2], Hastings [3] and Gardner and Smoller [4]), implicit function theorem and related functional-analytic techniques (Fife [5], Fujii et al. [6], Hale and Sakamoto [7,8]) differential inequalities (see for instance Chang and Howes [9]) and nonstandard analysis (Diener and Reeb [10]).