The Hilbert phenomenon in chaotic motions

It is proved that chaotic dynamical systems have D1-singularity. Due to such singularity and its periodicity in Hausdorff metric spaces, a chaotic dynamical system induces a Hilbert iterated system, which was discovered at the end of the 19th century. This implies that the most puzzling discovery in nature in the 20th century can be put into deep correspondence with that in thoughts in the 19th century. All this is verified by the numerical experiments of chaotic vibrations of a beam and a tension-slack oscillator.