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Energy harvesting systems based on oscillators aim to capture energy frommechanical oscillations and convert it into electrical energy. Widely extendedare those based on piezoelectric materials, whose dynamics are Hamiltoniansubmitted to different sources of dissipation: damping and coupling. Thesedissipations bring the system to low energy regimes, which is not desired inlong term as it diminishes the absorbed energy. To avoid or to minimize suchsituations, we propose that the coupling of two oscillators could benefit fromtheory of Arnold diffusion. Such phenomenon studies $O(1)$ energy variations inHamiltonian systems and hence could be very useful in energy harvestingapplications. This article is a first step towards this goal. We consider twopiezoelectric beams submitted to a small forcing and coupled through anelectric circuit. By considering the coupling, damping and forcing asperturbations, we prove that the unperturbed system possesses a $4$-dimensionalNormally Hyperbolic Invariant Manifold with $5$ and $4$-dimensional stable andunstable manifolds, respectively. These are locally unique after theperturbation. By means of the parameterization method, we numerically computeparameterizations of the perturbed manifold, its stable and unstable manifoldsand study its inner dynamics. We show evidence of homoclinic connections whenthe perturbation is switched on.

Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators