Transfers of energy in an infinite pendulum lattice

Abstract

In this talk we present a result about the existence of transfer of energy orbits in a chain of infinitely many weakly coupled pendulums. The dynamics of the chain is described by an infinite dimensional nearlyintegrable Hamiltonian system on a lattice of any dimension. We consider both nearest-neighbor interactions among the pendulums as well as long-range interactions with certain decay. In both cases, we construct solutions that move the energy of pendulums along any
prescribed path in the lattice. This can be seen as a result of topological instability. Indeed, the energies of the pendulums are constants of motion when they are all uncoupled. The weak coupling acts as a small perturbation and we show that, under appropriate initial conditions, this allows such energies to exhibit a drastic change at certain times. In dynamical systems theory this phenomenon of topological instability goes under the name of Arnold diffusion. The argument has been vastly studied, but, at the best of our knowledge, only in the finitedimensional context. Here we develop some geometric and functional tools to apply Arnold diffusion mechanisms to infinite dimensional Hamiltonian systems.