Ruling out closed orbits

Suppose we have a strong suspicion, based on numerical evidence or otherwise, that a particular system has no periodic solutions. How could we prove this In the last chapter we mentioned one method, based on index theory (see Examples 

5 and 6.8.6). Now we present three other ways of ruling out closed orbits. They are of limited applicability, but they re worth knowing about, in case you get lucky. 

Suppose the system can be written in the form x = -VV, for some continuously differentiable, single-valued scalar function V(x). Such a system is calied a gradient system with potential function V. 

Theorem 7.2.1 Closed orbits are impossible in gradient systems. 

Proof Suppose there were a closed orbit. We obtain a contradiction by considering the change in V after one circuit. On the one hand, AV = 0 since V is single-valued. But on the other hand. 

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The next section presents two examples of systems with limit cycles. In the first case, the limit cycle is obvious by inspection, but normally it s difficult to tell whether a given system has a limit cycle, or indeed any closed orbits, from the governing equations alone. Sections 7.2-7.4 present some techniques for ruling out closed orbits or for... 

The third method for ruling out closed orbits is based on Green s theorem, and is known as Dulac s criterion. 

Now that we know how to rule out closed orbits, we turn to the opposite task finding methods to establish that closed orbits exist in particular systems. The following theorem is one of the few results in this direction. It is also one of the key theoretical results in nonlinear dynamics, because it implies that chaos can t occur in the phase plane, as discussed brief y at the end of this section. 

An extended Hiickel MO calculation supports the assumptions made in the above analysis in that the three t2g orbitals are indeed close together in energy and remain nearly nonbonding metal-based d-orbitals. The detailed agreement is less satisfactory the SOMO is predicted to be primarily dx2 y2 with a small dxz admixture (hybrid 6 of Table 4.13), a result that can be ruled out from our analysis of the ESR results. The EHMO overlap matrix based on the X-ray structure suggests that the molecule is much closer to C2 symmetry than to Cs. If we accept that conclusion, then dxzjdxy hybridization is less likely than dyjdxy, as we tacitly assumed above. 

The photochemistry of tethered alkenes is more predictable than the nontethered situation. There is evidence of Ti-stacking for closely held moieties which presumably improves the orbital interactions between the alkenes. Exciplex formation is likely involved when the reacting groups are within 5 angstroms. Cornil et al. have shown that k systems can couple within this distance [11]. This exciplex could lead to a concerted cycloaddition from the excited state which would be consistent with the observed products. Although stepwise addition (see Sch. 3) cannot be ruled out even in these tethered singlet reactions. Ring closure must be very rapid if diradicals are involved, since no radical-trapped species have been found. 

The crystal structure of Sn(CH(SiMe3)2 (see Fig. 8), however, shows this interesting compound to be a centrosymmetric dimer (69). In fact the Sn—Sn distance of 2.765 A is close to that in hexaphenylditin (2.77 A) (70). Since the solid is diamagnetic, a conventional tin-tin single bond is ruled out. Lappert has suggested that the tin-tin interaction be described as a bent double bond with the lone pair on each tin being donated into the empty orbital on the adjacent tin atom, as shown in Fig. 9. 

In this section we discuss index theory, a method that provides global information about the phase portrait. It enables us to answer such questions as Must a closed trajectory always encircle a fixed point If so, what types of fixed points are permitted What types of fixed points can coalesce in bifurcations The method also yields information about the trajectories near higher-order fixed points. Finally, we can sometimes use index arguments to rule out the possibility of closed orbits in certain parts of the phase plane. 

Theorem 6.8.2 has many practical consequences. For instance, it implies that there is always at least one fixed point inside any closed orbit in the phase plane (as you may have noticed on your own). If there is only one fixed point inside, it cannot be a saddle point. Furthermore, Theorem 6.8.2 can sometimes be used to rule out the possible occurrence of closed trajectories, as seen in the following examples. 

Caldwell et al. have also reported a stereoselective [2 + 2] photocycloaddition where the major product was the most thermodynamically stable. Scheme 54 shows that the cyclobutane product with the two aryl groups trans to each other predominates, but it is not the exclusive product. The proposed intermediates include a 1,2-biradical, where the p-orbitals are perpendicular to each other, and a 1,4-biradical intermediate which has time to assiune the most stable conformation before closing. The 1,2-biradical intermediate is supported by rate studies and quenching data, but these studies are not conclusive [41a]. In addition, the possibility of involvement of an exciplex prior to cycloaddition cannot be ruled out based on the studies Caldwell et al. have reported. 

The studies of such closely related structures also ruled out any possible steric effects and the driving influence for reaction rate enhancement has to be seen in the oxygen atom in the y-allylic position (C6 of the Claisen system. Scheme 14). In previous reports [23], Carpenter and Burrows had developed a model to predict the influence of substituents on various pericyclic reactions based on Hiickel orbital energy calculations. According to this approach, a n-... 

As a first step, we have studied the influence of the nature of the functional on the charge-transfer character in the ground state of a cofacial TTF/TCNQ complex with the two molecules separated by 3.5 A. It is worth stressing that CASSCF and CAS-MRCI calculations, used as benchmark in the previous studies, rule out a complete electron transfer in the ground state [9, 10]. Only a partial charge transfer (lower than 0.15 lei) takes place, which validates the use of a mono-determinantal closed-shell approach such as DFT. However, the shift of the frontier orbitals correlated with the amount of HF exchange should affect the amount of charge transferred. 

Numerous asteroids and smaller debris also exist outside the belt, that is at distances greater than 3.2 AU as well as smaller than 2.2 AU. In particular, the Apollo, Amor, and Atem groups of asteroids are of concern because their orbits cross, or come close to, that of Earth. Smaller objects, less than a few kilometers in diameter, are difficult to detect from Earth, but a collision with one of them could have catastrophic consequences, even if it had a diameter of only a few hundred meters. In 1995, an asteroid about 5 km in length passed between Earth and the Moon it was not detected until days later. It is believed that a large asteroid impacted Earth 65 million years ago and was responsible for the extinction of the dinosaurs and many other species. It is quite possible that a similar event may have caused an even more devastating extinction of many forms of life nearly 250 million years ago (Ward et al, 2000). The possibility that catastrophic impacts may occur again cannot be ruled out (Gehrels, 1994). 

Calculations predict that the lowest state of PN has an open-shell electronic configuration." " The Salem-Rowland Rule for ISC promoted by spin-orbit coupling (SOC) predicts that singlet to triplet relaxation will have its maximum rate when the singlet state is closed-shell. This is the case with diaryl carbenes where the absolute rate constants of ISC are in the order of Michl has recently pointed out the importance of donor-... 

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