Nonlinear Systems & Linearization · Superchallenge problem

Traveling waves of a bistable reaction–diffusion equation lead to the system $u' = v,\; v' = -cv - u(1 - u)\left(u - \tfrac14\right)$. The equilibria $(1, 0)$ and $(0, 0)$ are both saddles, and there is exactly one value $c > 0$ for which a heteroclinic orbit runs from $(1, 0)$ to $(0, 0)$. Find $c$.