Three-dimensional torus

Chaos does not occur as long as the torus attraaor is stable. As a parameter of the system is varied, however, this attractor may go through a sequence of transformations that eventually render it unstable and lead to the possibility of chaotic behavior. An early suggestion for how this happens arose in the context of turbulent fluid flow and involved a cascade of Hopf bifurcations, each of which generate additional independent frequencies. Each additional frequency corresponds to an additional dimension in phase space the associated attractors are correspondingly higher dimensional tori so that, for example, two independent frequencies correspond to a two-dimensional torus (7 ), whereas three independent frequencies would correspond to a three-dimensional torus (T ). The Landau theory suggested that a cascade of Hopf bifurcations eventually accumulates at a particular value of the bifurcation parameter, at which point an infinity of modes becomes available to the system this would then correspond to chaos (i.e., turbulence). 

The assumption on U is not satisfied by some simple potentials. For example consider U(x,y) = which is completely independent of y and so places no restriction on that variable for constant energy. The condition on U means that it has a confining property. To ensure bounded solutions in systems that do not have a confining potential, we should add other assumptions, for example the assumption that the position vectors of each atom are restricted by periodic boundary conditions to lie on a three-dimensional torus (see Section 1.6). 

Consider a geodesic flow of a flat two-dimensional torus, that is, a torus with a locally Euclidean metric. This flow is integrable in the class of Bott integrals and obviously has no closed stable trajectories. By virtue of Proposition 2.1.2, we must have rank i(Q) 2. Indeed, the nonsingular surfaces Q are diffeomorphic here to a three-dimensional torus T, for which Hi T, Z) = Z 0 Z 0 Z. 

There has been some recent work on the use of spherical SOMs. Just as a SOM shaped as a ring is one-dimensional (each node only has neighbors to the left and right), so a spherical SOM resembles a torus and is two-dimensional (neighbors to the left and right, and also to the top and bottom, but not above and below), so a spherical SOM should be faster in execution than a genuinely three-dimensional SOM. 

The effect of sufficiently weak anharmonicities of the potential on this picture will be to distort the rectangle comprising the classical trajectories so that the motion occurs on a two-dimensional torus belonging to the three-dimensional constant energy subspace of the total four-dimensional phase space of the system [Arnold, 1978]. 

Finally, we mention that nonplanar graphs having one unavoidable intersection of edges can be represented on the surface of a torus without such an intersection. The nonplanar graph as well as the skeleton graph of the Mobius compound synthesized by Walba et dl (which is homeomorphic to /Ca ) can be drawn on a torus surface without intersections. The surface of the torus is also important for other unusual chemical compounds The skeletons of catenanes, rotaxanes and knots cannot be embedded in the surface of a three-dimensional sphere, but in that of a torus. 

Figure 7 (Top left) Torus of constant-energy vibrational motion in two uncoupled degrees of freedom, projected upon the three-dimensional space (9, q, pz). (Below right) Three nested tori, sliced to reveal their foliated structure.

Cutting the tori along these circles, we obtain two rings out of each torus. Gluing the round handle and considering the boundary of the obtained three-dimensional manifold, we get one torus as the upper component of the boundary. 

Fig. 2.17. Illustration of the periodic boundary conditions, (a) A one-dimensional system of spheres (rods). The periodic boundary condition is obtained by identifying the end A with A. The result is a closed circle, (b) A two-dimensional system of spheres (disks). The periodic boundary condition is obtained by identifying the edges A with A and B with B, The result is a torus, (c) A three-dimensional system of spheres. The periodic boundary condition is obtained by identifying opposite faces, such as and A, with each other. No pictorial description of the resulting system is possible.

For bulk three-dimensional solids, g is equivalent to the number of cuts required to transform a solid structure into a structure topologically equivalent to a sphere (for instance, g = 0 for a polygonal sphere such as or C70, and g = 1 for a torus). Suppose, further, that the object is formed of polygons having different (0 number of sides. The total number of faces (F) is then... 

Kohonen map of this molecular electrostatic potential. The Kohonen projection is made onto the surface of a torus to produce a map without beginning and without end just as the surface of a molecule. The positive electrostatic potential (dark blue) cannot be seen in three-dimensional model because it is hidden on the opposite side... 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

It is necessary to emphasize one principal peculiarity of the copolymerization dynamics which arises under the transition from the three-component to the four-component systems. While the attractors of the former systems are only SPs and limit cycles (see Fig. 5), for the latter ones we can also expect the realization of other more complex attractors [202]. Two-dimensional surfaces of torus on which the system accomplishes the complex oscillations (which are superpositions of the two simple oscillations with different periods) ate regarded to be trivial examples of such attractors. Other similar attractors are fitted by the superpositions of few simple oscillations, the number of which is arbitrary. And, finally, the most complicated type of dynamic behavior of the system when m 4 is fitted by chaotic oscillations [16], for which a so-called strange attractor is believed to be a mathematical image [206]. 

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