On the destruction of three-dimensional tori

We have studied the direct transition from three-frequency oscillations on T-3 to chaos. The onset of chaos is analyzed where the torus loses its smoothness in the neighbourhood of a resonance. We have proposed two methods to reveal the destruction of the torus T-3. The first method is based on the iteration of parts of the attractor to demonstrate stretching and folding. The second one is related to the evolution of the critical surfaces of noninvertible maps, which are generalizations of the critical points in one-dimensional maps. With both techniques we can show that the torus is indeed broken at the onset of chaos. An important advantage of these methods is that they can be easily applied even in cases where the expansion is much smaller than the contraction. Note that the estimation of the box dimension of such attractors yields integer values, because the broken structure of the attractor is too tiny to be detected by estimating the capacity.