I am trying to understand the persistence of a heteroclinic orbit $(u^*,v^*,c^*)$ in the FitzHugh-Nagumo equation. I use geometric singular perturbation theory as described by Jones in "Geometric Singular Perturbation Theory" in "Dynamical Systems: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 13 - 22, 1994 (Lecture Notes in Mathematics, 1609)". I understand that the key point is the transversality of the stable manifold $W^s(\{1\}\times I)$ and the unstable manifold $W^u(\{a\}\times I)$ of the reduced sytem $$\begin{aligned} u'&=v\\ v'&=-cv-f(u)\\ c'&=0 \end{aligned}$$ for $f(u)=u(u-a)(1-u)$ and $I$ a compact interval containing the value $c=c^*$ for which the heteroclinic orbit exists. However, I don't understand how he proves the transversality. On pages 90-92 he first defines the intersection of the stable and unstable manifolds with a plane normal to the $u$-axis as graphs of some differentiable functions $h^+(c)$ and $h^-(c)$, respectively. At this point I do not understand why these functions have to exist, since the manifolds $W^s,W^u$ may be perpendicular to the $c$-axis in any point between the ends of the orbit.
Using these functions $h^\pm$ he finds tangent vectors $\eta^\pm = (0,\frac{\partial h^\pm(c^*)}{\partial c},1)$ and $\eta=(v,-cu-f(u),0)$ to the orbit in any point and deduces some information about their flow $\Phi_t$ under the equation linearized in the heteroclinic orbit. It remains to be shown that $\frac{\partial h^+(c^*)}{\partial c}\neq \frac{\partial h^-(c^*)}{\partial c}$. In particular, he shows that $$du\wedge dv (\eta^\pm,\eta)=v\frac{\partial h^\pm(c^*)}{\partial c}\text{ (I think he misses a minus here)}$$ and that the linearized flow satisfies $$du\wedge dc(\Phi_t(\eta^\pm),\Phi_t(\eta))=-v^*(t).$$ He infers that the function $\omega(t)=du\wedge dv (\Phi_t(\eta^\pm),\Phi_t(\eta))$ must satisfy the differential equation $$\omega'=-c^*\omega -{v^*}^2$$ and therefore be of the form $$\omega(t) = e^{-c^*t}\int_{-\infty}^te^{c^*\tau}{v^*(\tau)}^2d\tau+Ke^{-c^*t}.$$ I think I can follow his arguments until this point, but then he says it's an excercise to show that $\lim_{t\to-\infty}e^{c^*t}\omega(t)=0$. I have no idea how to show this, do you have any tips for that? From that he then follows that $K=0$ and that $\frac{\partial h^-(c^*)}{\partial c}>0$. He then says that from a similar calculation we can follow that $\frac{\partial h^+(c^*)}{\partial c}<0$, thus proving the transversality. However, I also don't know how such a similar calculation would look like.
If you have any ideas, your help is greatly appreciated.