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The aim of this note is to present the recent results in  where we provide the existence of solutions of some nonlinear resonant PDEs on T2 exchanging energy among Fourier modes in a "chaotic-like" way. We say that a transition of energy is "chaotic-like" if either the choice of activated modes or the time spent in each transfer can be chosen randomly. We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations. The key point of the construction of the special solutions is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form of those equations.
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Filippo Giuliani, Marcel Guardia, Pau Martin, Stefano Pasquali, Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 149–166DOI 10.4171/RLM/931