  Classical results from Sturm-Liouville theory state that the number of
unstable eigenvalues of a scalar, second-order linear operator is equal to the
number of associated conjugate points. Recent work has extended these results
to a much more general setting, thus allowing for spectral stability of
nonlinear waves in a variety of contexts to be determined by counting conjugate
points. However, in practice, it is not yet clear whether it is easier to
compute conjugate points than to just directly count unstable eigenvalues. We
address this issue by developing a framework for the computation of conjugate
points using validated numerics. Moreover, we apply our method to a
parameter-dependent system of bistable equations and show that there exist both
stable and unstable standing fronts. This application can be seen as
complimentary to the classical result via Sturm-Louiville theory that in scalar
reaction-diffusion equations pulses are unstable whereas fronts are stable, and
to the more recent result of "Instability of pulses in gradient
reaction-diffusion systems: a symplectic approach," by Beck et. al., that
symmetric pulses in reaction-diffusion systems with gradient nonlinearity are
also necessarily unstable.
