  Delta Scuti ($\delta$ Sct) variables are intermediate mass stars that lie at
the intersection of the main sequence and the instability strip on the
Hertzsprung-Russel diagram. Various lines of evidence indicate that nonlinear
mode interactions shape their oscillation spectra, including the particularly
compelling detection of resonantly interacting mode triplets in the $\delta$
Sct star KIC 8054146. Motivated by these observations, we use the theory of
three-mode coupling to study the strength and prevalence of nonlinear mode
interactions in fourteen $\delta$ Sct models that span the instability strip.
For each model, we calculate the frequency detunings and nonlinear coupling
strengths of $\sim 10^4$ unique combinations of mode triplets. We find that all
the models contain at least $\sim 100$ well-coupled triplets whose detunings
and coupling strengths are consistent with the triplets identified in KIC
8054146. Our results suggest that resonant mode interactions can be significant
in $\delta$ Sct stars and may explain why many exhibit rapid changes in
amplitude and oscillation period.
