I am currently reading Geometric Singular Perturbation Theory (exceptionally helpful by the way), and it talks about tracking stable and unstable manifolds with regards to traveling waves.

Consider the Fitzhugh Nagumo Model:

\begin{align} u_t&=u_{xx}+f(u)-w\\ w_t&=\epsilon(u-\gamma w), \end{align} where $f(u)=u(u-a)(1-u)$, $0<a<1/2$. Letting $\epsilon=0$ and $u(x,t)=U(\xi)$ (where $\xi=x-ct$), the system reduces to the scalar equation: \begin{align} u'&=v\\ v'&=-cv-f(u)\\ c'&=0 \end{align}

It has been established that there exists a unique $c=c^* >0$ such that a heteroclinical orbits connects $0=U(-\infty)$ to $1=U(+\infty)$. Anyways, here's the part I'm confused about:

It can be easily shown that $W^-$, the center-unstable manifold of the curve $\{(0,0,c): c \text{ near } c^*\}$ and $W^+$, the center-stable manifold of the curve $\{(1,0,c): c \text{ near } c^*\}$ are both two dimensional. He claims that by Implicit Function Theorem, we can say at the $u=a$ plane (so $f(u)=0)$ we may write the unstable and stable manifolds as graphs $v=v^-(c)$ and $v=v^+(c)$ respectively.

1) I understand the intuition behind this fact: Both are two dimensional, the tangent space of the system provides one dimension to both manifolds, leaving one dimension in the $(v,c)$ plane, so it must be a curve. Is this reasoning correct?

2) How exactly does one arrive at the fact using Implicit Function Theorem?

Thank you.


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