Wrinkled 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. 

Figure 28 Definition of the angle 6 from the Poincare section of a torus attractor derived from experimental data. The index labels the order with which points appear in this section as the trajectory winds its way over the surface of the torus. This definition can be generalized to a wrinkled or fractal torus.

Chaos may also occur as a consequence of the destruction of a two-torus that characterizes a quasiperiodic regime with two incommensurate frequencies. Quasiperiodicity in chemistry was recently discovered in experiments on the BZ reaction, and chaos was reached through the development of wrinkles on the torus (ROUX and ROSSI, this volume [52]). ARGOUL et al, [53] (in this volume) have proposed a tentative interpretation of the experiments. 

Another predicted route to turbulence is through the wrinkling of a two-torus. This too has been observed in experiments on the BZ reaction. 

The circled zones in figure 4 represent, at the same scale, the corresponding Poincare sections of the limit cycle that exists prior to the bifurcation toward the torus. The aireas of these dashed zones stand for the e3q>erimental scatter of the points. It is isotropic almost everywhere except in the part of the limit cycle which corresponds to the stretching of the wrinkle. Furthermore, we must remark that the limit cycle appears to lie on the surface of the torus. This is cin indication that the bifurcation leading to the torus is of a saddle-node type rather than of a Hopf type (in this latter case the limit cycle should be inside the torus). This character is also confirmed by the abruptness of the transition eind the absence of hysteresis. 

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