Suppose the reaction-diffusion system Suppose the dynamical system \begin{align*} v_t &= f(v, w) \\ w_t &= g(v, w) \end{align*} where $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$, has an asymptotically stable fixed point at $(v^*, w^*)$ and consider the spatially extended system, \begin{align*} v_t &= D v_{xx} + f(v, w) \\ w_t &= g(v, w) \end{align*} put into traveling wave coordinates (make the change of variable $\xi = x + ct$ to get a system ODEs) \begin{align*} v_{\xi} &= u \\ u_{\xi} &= D^{-1}(cu - f(v, w)) \\ w_{\xi} &= \frac{1}{c}g(v, w) \end{align*}
I'd like to show that the system of ODEs will always have a saddle structure at the point $(v^*,\ 0,\ w^*)$. I was thinking of trying to use an argument involving a rank-1 update to the jacobian of the system linearized about the saddle-point, but have fallen short. Are there any references with a proof of this? I've seen it done for specific models like Fitzhugh-Nagumo and Morris-Lecar but would like to show it in general.