On the “traveling pulses” of the limit of the FitzHugh–Nagumo equation when ɛ→0

A solution (u(s),v(s)) of the differential system u^′=v,v^′=−cv−u(u−a)(1−u)+w,w^′=−(ɛ/c)(u−γw).with a,c,ɛ∈R such that (u(s),v(s))→(0,0) when s→±∞ is a traveling pulse of the FitzHugh–Nagumo equation. The limit of this differential system when ɛ→0 gives rise to the polynomial differential system u^′=v,v^′=−cv−u(u−a)(1−u)+w,where now a,c,w∈R. We give the complete description of its phase portraits in the Poincaré disc (i.e. in the compactification of R² adding the circle S¹ of the infinity) modulo topological equivalence.