Double tori

Wang J, Sorensen P G and Flynne F 1994 Transient period doublings, torus osoillations and ohaos in a olosed ohemioal system J. Phys. Chem. 98 725-7... 

The molecular structure of [Nae(H20)eBie(cr-CDH.i2)2] 137 in crystals of 137 47 H2O in a view through the double torus (137a) and in a view onto the double torus (137b). Bi atoms (violet), Na atoms (yellow). Hydrogen atoms omitted for clarity. Dashed lines hydrogen bonds... 

Figure 20. Cylindrical polyhexes which, by bridging two holes in the surface of a toroidal polyhcx. can form a double torus by inserting or gluing edges. A hole of type A can be joined by gluing a cylinder of type 1, and B by connecting with edges to one of type II, whereas most holes (e.g., C) are more complicated. (Note that none of the results have a purely polyhex surface.)...

The doubly toroidal C6o isomer mentioned had 24 hexagonal faces and 4 of 9 vertices each, totaling 28 faces, and conforming with equation (7), Section 8.4. A quite different kind of double torus was recently suggested by Klein and Liu. This looks rather like a cotton reel and consists of two separate tori connected by a cylinder (Figure 21). It involves the merging of one lattice sheet onto the surface of another (not passing... 

Figure 21. Another kind of "double torus in the. sense of being two separate tori connected with a single (cylindrical) lattice sheet. (From Ref. 21a.)...

The stability characteristics of electrodynamic balances with electrode configurations other than bihyperboloidal are affected by the different electrode geometry. Once the electrical fields are determined for the geometry in question, the stability characteristics can be established quantitatively. Muller (1960) developed expressions for the electrical fields for several configurations, including Straubel s disk and torus system, and Davis et al. (1990) analyzed the double-ring which is discussed below. 

The most important characteristic in our test cases, however, is that within the 1/1 and the 2/1 resonance horns the torus will break as FA increases. In all models this happens when the unstable source period 1 that existed within the torus hits the saddle-periodic trajectories that lie on the torus. This occurs through a saddle-node bifurcation in the 1/1 resonance horn [Fig. 8(d)], and through an unstable period doubling in the 2/1 resonance [Fig. 8(c)]. After these bifurcations the basic structure of the torus has collapsed, and we are left only with the stable entrained periodic trajectories. 

The three standard local codimensional-one bifurcations are the saddle-node, Hopf, and period doubling bifurcations and several have been continued numerically for this model and appear in figure 2. We have chosen not to show the curves of focus-node transitions because they do not represent any changes in stability, only changes in the approach to the steady behaviour. The saddle-node bifurcations that occur during phase locking of the torus at low amplitudes continue upward and either close upon themselves as in the case of the period 3 resonance horns or the terminate in some codimension-two bifurcation. 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

Atmospheric effects of large-scale TNT expins have also been studied in depth both practically and theoretically. Factors considered include pressure and impulse effects, decay characteristics and travel and duration times, all as a function of distance, and for both free-field and reflection situations (Refs 3,9,15,16, 17,24,32, 33,34,35,36,44, 53,75,76,115 116). A distinction is made between the blast area dose to the source, comprising air and the products of expln, and that farther away involving air only (Ref 53). Double-burst conditions (fireball and shock wave interaction, and torus formation) have been studied (Ref 149), as have also the dynamics of dust formation and motion (Refs 25,26 117). Performance tests were run on a naval blast valve (Ref 92), and on aircraft wing panels (Ref 4)... 

From a physical point of view, the rhythmic phenomena are related to the fact that biological systems are maintained under far-from-equilibrium conditions through a continuous dissipation of energy [23]. However, non-equilibrium conditions can also give rise to more complicated behaviors. Chaotic dynamics, for instance, can arise either as a regular rhythmic process is destabilized and develops through a cascade of period-doubling bifurcations [24], by torus destruction in connection with the interaction of two or more rhythms, or via different types of intermittency... 

Fig. 3.7. (Left) Schematic image of a torus. (Right) Double-logarithmic plot of the torus size, average radius R, and thickness r vs. chain length L (3.4) with parameters 7a2/T = 4 and l/a = 15. Also shown are the results for a charged semiflexible chain (cf. Sect. 3.3.4 and [32] for more details)...

Higher degrees of dynamic complexity are possible, including period doubling, quasi periodicity (torus) and chaos. Phase plane plots are not the best means of investigating these complex dynamics for in... 

A torus or anchor ring, drawn in Fig. 1.3, is the approximate shape of a donut or bagel. The radii R and r refer, respectively, to the circle through the center of the torus and the circle made by a cross-sectional cut. Generally, to determine the area and volume of a surface of revolution, it is necessary to evaluate double or triple integrals. However, long before calculus was invented. Pappus of Alexandria (ca. Third Century A.D.) proposed two theorems that can give the same results much more directly. 

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. 

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