  We deal with dynamical systems on complex lattices possessing chains of
non-transversal heteroclinic connections between several periodic orbits. The
systems we consider are inspired by the so-called \emph{toy model systems}
(TMS) used to prove the existence of energy transfer from low to high
frequencies in the \emph{nonlinear cubic Schr\"odinger equation} (NLS) or
generalizations. Using the geometric properties of the complex projective space
as a base space, we generate in a natural way collections of such systems
containing this type of chains, both in the Hamiltonian and in the
non-Hamiltonian setting. On the other hand, we characterize the property of
block diagonal dynamics along the heteroclinic connections that allows these
chains to be shadowed, a property which in general only holds for transversal
heteroclinic connections. Due to the lack of transversality, only finite chains
are shadowed, since there is a dropping dimensions mechanism in the evolution
of any disk close to them. The main shadowing technical tool used in our work
is the notion of covering relations as introduced by one of the authors.
