In this talk we consider two PDEs on the 2-dimensional torus,the Hartree and Beam equation,and we show the existence of solutions which are essentially Fourier supported on 8 resonant modes and have a non trivial dynamics that involves exchanges of energy in a chaotic fashion. More precisely,at certain time scales some modes oscillate between two values and the exchange times can be chosen randomly within a certain range. This behaviour is due to the presence of an invariant horseshoe-like set for the truncated Birkhoff normal form. This is a work in progress with M. Guardia, P.Martin, S. Pasquali.
org: PROCESI Michela