  We study the chaos of travelling waves (TW) in unidirectional chains of
bistable maps. Previous numerical results suggested that this property is
selective, {\sl viz.}\ given the parameters, there is at most a single
(non-trivial) velocity for which the corresponding set of wave profiles has
positive topological entropy. However, mathematical proofs have remained
elusive, in particular because the related symbolic dynamics involves entire
past sequences. Here, we consider instead inite (short) rank approximations for
which the symbolic dynamics has finite memory. For every possible velocity, we
compute the existence domains of all possible finite type subshifts of TW with
positive entropy. In all examples, chaos of TW turns out to be selective,
indeed.
