  We consider the Kirchhoff equation on tori of any dimension and we construct
solutions whose Sobolev norms oscillates in a chaotic way on certain long time
scales. The chaoticity is encoded in the time between oscillations of the norm,
which can be chosen in any prescribed way. This phenomenon, that we name as
effective chaos (it occurs over a long, but finite, time scale), is 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 reductions, this model behaves as two
weakly coupled pendulums.
