  Energy harvesting systems based on oscillators aim to capture energy from
mechanical oscillations and convert it into electrical energy. Widely extended
are those based on piezoelectric materials, whose dynamics are Hamiltonian
submitted to different sources of dissipation: damping and coupling. These
dissipations bring the system to low energy regimes, which is not desired in
long term as it diminishes the absorbed energy. To avoid or to minimize such
situations, we propose that the coupling of two oscillators could benefit from
theory of Arnold diffusion. Such phenomenon studies $O(1)$ energy variations in
Hamiltonian systems and hence could be very useful in energy harvesting
applications. This article is a first step towards this goal. We consider two
piezoelectric beams submitted to a small forcing and coupled through an
electric circuit. By considering the coupling, damping and forcing as
perturbations, we prove that the unperturbed system possesses a $4$-dimensional
Normally Hyperbolic Invariant Manifold with $5$ and $4$-dimensional stable and
unstable manifolds, respectively. These are locally unique after the
perturbation. By means of the parameterization method, we numerically compute
parameterizations of the perturbed manifold, its stable and unstable manifolds
and study its inner dynamics. We show evidence of homoclinic connections when
the perturbation is switched on.
