Effective chaos for the Kirchhoff equation on tori

We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillate in a chaotic way on certain long timescales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, which we name effective chaos (it occurs over a long, but finite, timescale), is a consequence of the existence of symbolic dynamics for an effective system. Since the first-order resonant dynamics has been proved to be essentially stable, we need to perform a second-order analysis to find an effective model displaying chaotic dynamics. More precisely, after some nontrivial reductions, this model behaves as two weakly coupled pendulums.