How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))?
I started by finding the stable manifold of the stable node and the unstable manifold of the saddle point.
Next, I want to show that their intersection is not empty and so there must be a trajectory that starts at s and end at n. And therefore, a heteroclinic orbit. But the problem is that the stable manifold only "covers" a neighborhood of n. Same for the unstable manifold.
So how do we figure out the neighborhoods I guess? Or is there a better way to do this than using the stable and unstable manifolds?
Edit: This is the system of ODEs I'm dealing with. $$U' = V\\ V' = -\frac{c}{D}V -\frac{\mu}{D} U (1-U)\\$$ All of the parameters are positive. and $c>\sqrt{4\mu D}$. So we have two equilibria: A stable node: (U,V) = (0,0). And a saddle point: (U,V) = (1,0).
Thanks