  We develop a stability index for the travelling waves of non-linear reaction
diffusion equations using the geometric phase induced on the Hopf bundle
$S^{2n-1} \subset \mathbb{C}^n$. This can be viewed as an alternative
formulation of the winding number calculation of the Evans function, whose
zeroes correspond to the eigenvalues of the linearization of reaction diffusion
operators about the wave. The stability of a travelling wave can be determined
by the existence of eigenvalues of positive real part for the linear operator.
Our method of geometric phase for locating and counting eigenvalues is inspired
by the numerical results in Way's Dynamics in the Hopf bundle, the geometric
phase and implications for dynamical systems. We provide a detailed proof of
the relationship between the phase and eigenvalues for dynamical systems
defined on $\mathbb{C}^2$ and sketch the proof of the method of geometric phase
for $\mathbb{C}^n$ and its generalization to boundary-value problems.
Implementing the numerical method, modified from Way's work, we conclude with
open questions inspired from the results.
