  A Koopman decomposition is a powerful method of analysis for fluid flows
leading to an apparently linear description of nonlinear dynamics in which the
flow is expressed as a superposition of fixed spatial structures with
exponential time dependence. Attempting a Koopman decomposition is simple in
practice due to a connection with Dynamic Mode Decomposition (DMD). However,
there are non-trivial requirements for the Koopman decomposition and DMD to
overlap which mean it is often difficult to establish whether the latter is
truly approximating the former. Here, we focus on nonlinear systems containing
multiple simple invariant solutions where it is unclear how to construct a
consistent Koopman decomposition, or how DMD might be applied to locate these
solutions. First, we derive a Koopman decomposition for a heteroclinic
connection in a Stuart-Landau equation revealing two possible expansions. The
expansions are centred about the two fixed points of the equation and extend
beyond their linear subspaces before breaking down at a crossover point in
state space. Well-designed DMD can extract the two expansions provided that the
time window does not contain this crossover point. We then apply DMD to the
Navier-Stokes equations near to a heteroclinic connection in low-Reynolds
number ($Re=O(100)$) plane Couette flow where there are multiple simple
invariant solutions beyond the constant shear basic state. This reveals as many
different Koopman decompositions as simple invariant solutions present and
again indicates the existence of crossover points between the expansions in
state space. Again, DMD can extract these expansions only if it does not
include a crossover point.
