Broken torus

Whereas the fractal torus is difficult to distinguish from the wrinkled torus, the broken torus (stage 4) is immediately recognizable from its surface of section. The transition from wrinkled to fractal torus can, however, be clearly seen in the associated circle map. The circle map develops an inflection point (see Figure 34) at the transition from wrinkled to fractal torus. The existence of an inflection point means that the circle map is no longer invertible, that is, the circle map cannot be derived from a true two-dimensional torus. It also means that chaotic dynamics are now possible. The transition from stage 2 to stage 3 heralds the death of the two-dimensional torus and the transition to the possibility of chaotic dynamics. 

Davis and Gray also demonstrated the existence of a series of intramolecular energy transfer bottlenecks, each corresponding to the breakup of a KAM torus. For example, for I2 in the vibrational state v = 20 they found intramolecular bottlenecks associated with frequency ratios equal to (3 + g) and up. However, Davis and Gray found that the last golden mean torus to be broken up is the most effective bottleneck to intramolecular energy transfer and is therefore... 

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