Fractal torus

Is shown for a high lying eigenstate of the Henon-Helles problem. This — possibly fractal — torus nevertheless gave an excellent semlclasslcal eigenvalue. 

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.

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. 

This example shows that mixed-mode oscillations, while arising from a torus attractor that bifurcates to a fractal torus, give rise to chaos via the familiar period-doubling cascade in which the period becomes infinite and the chaotic orbit consists of an infinite number of unstable periodic orbits. Mixedmode oscillations have been found experimentally in the Belousov-Zhabotin-skii (BZ) reaction 2.84 and other chemical oscillators and in electrochemical systems, as well. Studies of a three-variable autocatalator model have also provided insights into the relationship between period-doubling and mixedmode sequences. Whereas experiments on the peroxidase-oxidase reaction have not been carried out to determine whether the route to chaos exemplified by the DOP model occurs experimentally, the DOP simulations exhibit a route to chaos that is probably widespread in the realm of nonlinear systems and is, therefore, quite possible in the peroxidase reaction, as well. 

Chaotic behavior in nonlinear dissipative systems is characterized by the existence of a new type of attractor, the strange attractor. The name comes from the unusual dimensionality assigned to it. A steady state attractor is a point in phase space, whereas a limit cycle attractor is a closed curve. The steady state attractor, thus, has a dimension of zero in phase space, whereas the limit cycle has a dimension of one. A torus is an example of a two-dimensional attractor because trajectories attracted to it wind around over its two-dimensional surface. A strange attractor is not easily characterized in terms of an integer dimension but is, perhaps surprisingly, best described in terms of a fractional dimension. The strange attractor is, in fart, a fractal object in phase space. The science of fractal objects is, as we will see, intimately connected to that of nonlinear dynamics and chaos. 

The trapping process was simulated using a torus centered around each repeat unit in the cyclic.Any empty torus was considered a pathway for a chain of specified diameter to thread and then incarcerate the cyclic once the end-linking process has been completed. Simulations were consistent with experimental trapping efficiencies. It is possible to interpret these experimental results in terms of a power law for the trapping probabilities and fractal cross sections for the PDMS chains. 

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