Tori Equation

Figure 54. The application of Tori equation calculations to the prediction of H-3 coupling constants of 5a-10a-cw-normal-clerodane 3,4-a- and jS-epoxides 378).

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

Exercise 4.10 Consider the (topological) two-torus in and outer radius R. An equation for this two-torus is... 

The pneumatic drying model was solved numerically for the drying processes of sand particles. The numerical procedure includes discretization of the calculation domain into torus-shaped final volumes, and solving the model equations by implementation of the semi-implicit method for pressure-linked equations (SIMPLE) algorithm [16]. The numerical procedure also implemented the Interphase Slip Algorithm (IPSA) of [17] in order to account the various coupling between the phases. The simulation stopped when the moisture content of a particle falls to a predefined value or when the flow reaches the exit of the pneumatic dryer. 

So, apart from the regular behavior, which is either steady-state, periodic, or quasi-periodic behavior (trajectory on a torus, Figure 3.2), some dynamic systems exhibit chaotic behavior, i.e., trajectories follow complicated aperiodic patterns that resemble randomness. Necessary but not sufficient conditions in order for chaotic behavior to take place in a system described by differential equations are that it must have dimension at least 3, and it must contain nonlinear terms. However, a system of three nonlinear differential equations need not exhibit chaotic behavior. This kind of behavior may not take place at all, and when it does, it usually occurs only for a specific range of the system s control parameters 9. 

The equation applies only to orientable surfaces, those with distinct sides. This excludes one-sided surfaces, such as the Mobius strip.) Thus, a sphere has genus zero. A torus (or a sphere with one handle) has genus one, and so on. 

In this case the system is called integrable. From the first equation we have /, = const, for all i that is, /, are constants of motion. Thus / , (/) = 0//o/8/,- are also constants, and we have

phase space of the system is on a regular orbit (torus). 

Do you see why this knot corresponds to p = 3, <7 = 27 Follow the knotted trajectory in Figure 8.6.5, and count the number of revolutions made by during the time that 0, makes one revolution, where 0, is latitude and 0 is longitude. Starting on the outer equator, the trajectory moves onto the top surface, dives into the hole, travels along the bottom surface, and then reappears on the outer equator, two-thirds of the way around the torus. Thus 62 makes two-thirds of a revolution while 0 makes one revolution hence p = 3, q = 2. 

Equations (3.33) and (3.34) make a succinct statement about quantum-classical correspondence. Specilically, in the classical limit, stationary quantum eigendistributions correspond to stationary classical eigendistributions at the related quantized action. Secondly, nonstationary quantum eigendistributions pD>m(Ij,0) correspond to nonstationary classical eigendistributions. These consist of a nonuniform distribution on a torus at intermediate action I(m+D)/2 with the nonuniformity determined by the difference (n — m) times the angles on this torus.59 Thus, from this viewpoint there is a direct link between eigendistributions in quantum and classical mechanics in the h - 0 limit. 

The elimination of secular terms from the power series expansion of the solution is achieved by the method of Lindstedt. The underlying idea is to pick a fixed frequency p, and to look for a quasi-periodic solution with basic frequencies /i and v. This is actually the same thing as looking for a quasi-periodic orbit on an invariant 2-dimensional torus. The process of solution is the following. Write the Duffing s equation as... 

What is not immediately evident from the discussion above is how to write the equation for the Invariant torus. This is hidden in the discussion of Section 3.1. We should recall that the Lie series actually defines a coordinate transformation, although thanks to the formula (4) we can perform the whole normalization procedure without even mentioning it. Actually, denoting by p(s qt s > the coordinates that give the Hamiltonian the normal form up to order s, i.e., the form analogous to that represented in the diagram of H(s above, we can calculate the transformation in explicit form as... 

P. Mendes. A metric property of Cherry vector fields on the torus. J. Differ. Equations, 89(2) 305-316, 1991. 

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 33 Evolution of a torus attractor found in simulations with the DOP model of the peroxidase-oxidase reaction, equations [115]. Parameter values used are k2 = 1250, fea = 0.046875, = 1. 104, = 0.001,...

The membranous SCC duct is modeled as a section of a rigid torus filled with an incompressible Newtonian fluid. The governing equations of motion for the fluid are developed from the Navier-Stokes equation. Refer to the nomenclature section for definition of aU variables, and Figure 64.5 for a cross section of the SCC and membranous utricle sack. 

For a smectic-A that is confined in the SEA crossed cylinder geometry, there is a competition between the homeotropic alignment on mica and the tendency to form layers of equal thickness d. As a result, dislocations must arise (Fig. 3.18, inset). As the local geometry aroimd the contact point is equivalent to a sphere-plane geometry, the loops are expected to be circular and centered on the contact point. Consider thus an array of torus-like cells, coaxial to the loops (Fig. 3.18, inset). Each cell is defined by an inner radius r, corresponding to a thickness h ri) = Uid, and an outer radius rj+i, with h ri+i) = (rij - - l)d, and contains a circular dislocation loop of radius pi- The cells are independent, because the strain patterns produced by the dislocations decays exponentially outside a parabola of equation = z, where A ... 

Assume that the system (2.1) possesses a sufficiently smooth quasiperiodic solution jc (r) with frequency basis co. Then there exists a function (f) given on the torus such that A (r) = u (o>r) this function is a solution of the equation... 

Let us now find under what conditions (1.7) possesses an invariant torus. Consider the system of equations... 

We now apply Theorem 6.4. to prove the dieorem on the reducibility of the system of difference equations on a torus. 

Chapter 6 contains the investigation of invariant toroidal sets for the systems of difference equations with quasiperiodic coefficients. Here, as we see this, the interesting result is obtained, namely, the theorem on existence of the continuous torus is proved for the discrete dynamical system. In this chapter, we also present a method which enables us to construct the toroidal set in the form of a uniformly convergent sequence of toroidal sets and study the behavior of trajectories of a nonlinear system of difference equations on the torus and in its vicinity. 

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