Invariant manifolds in energy harvesting coupled oscillators

CDSNS Colloquium
Tuesday, July 25, 2017 - 11:00
1 hour (actually 50 minutes)
Skiles 005
Department of Applied Mathematics and Computer Science, Technical University of Denmark
When perturbed with a small periodic forcing, two (or more)  coupledconservative oscillators can exhibit instabilities: trajectories thatbecome unstable while accumulating ``unbounded'' energy from thesource.  This is known as Arnold diffusion, and has been traditionallyapplied to celestial mechanics, for example to study the stability ofthe solar system or to explain the Kirkwood gaps in the asteroid belt.However, such phenomenon could be extremely useful in energyharvesting systems as well, whose aim is precisely to capture as muchenergy as possible from a source.In this talk we will show a first step towards the application ofArnold diffusion theory in energy harvesting systems. We will consideran energy harvesting system based on two piezoelectric oscillators.When forced to oscillate, for instance when driven by a small periodicvibration, such oscillators create an electrical current which chargesan accumulator (a capacitor or a battery). Unfortunately, suchoscillators are not conservative, as they are not perfectly elastic(they exhibit damping).We will discuss the persistence of normally hyperbolic invariantmanifolds, which play a crucial role in the diffusing mechanisms. Bymeans of the parameterization method, we will compute such manifoldsand their associated stable and unstable manifolds. We will alsodiscuss the Melnikov method to obtain sufficient conditions for theexistence of homoclinic intersections.