Symmetric stretch orbit

The theory of bifurcations shows that the different types of bifurcations can be described in terms of normal forms, which represent local expansions of the dynamics around the bifurcating periodic orbit [19, 32, 49]. The purpose of the above mapping is to describe the successive bifurcations of the symmetric-stretch periodic orbit, starting from low energies above the saddle point. Appropriate truncation of the Taylor series of the potential v(q) around <7 = 0, which corresponds to the location of the symmetric-stretch orbit, provides us with the normal forms of the bifurcations [144], The bifurcations relevant for the dissociation dynamics under discussion can be described by truncating at the sixth order in q,... 

As the energy increases in the interval E < E < Ea, the orbits Y and 2 progressively shift toward the symmetric-stretch orbit 0 and merge at the subcritical antipitchfork bifurcation. Just below this bifurcation, 1 and 2 are elliptic while 0 is still hyperbolic (without reflection). Between and Ea, the periodic orbits 1 and 2 may either remain of elliptic type or become hyperbolic in the energy interval [ , "1 such that E < Edd> < Edd < Ea. 

The eigenfunctions associated with the resonances have been obtained via wavepacket propagation. They appear to be localized along the symmetric-stretch periodic orbit 0, with a number of nodes equal to n and even under the exchange of iodine nuclei. Due to the relative stability of the symmetric-stretch orbit, we have thus here a system where the hypothesis of the orbit 12 representing the RPO, that is, resonant periodic orbit, does not hold. 

In the following, we will characterize the different dynamical regimes for CO2 on the LEPS surface. The scenario in a general way resembles the one for Hgl2. However, the initial bifurcations are of the second type described in Section IV.C.2 that is, the symmetric-stretch orbit undergoes a subcritical... 

The period of the anti-symmetric stretch periodic trajectory does not correspond, however, to any of the three recurrences we see in Figure 8.4. This is not at all surprising in order to come back to the FC region, which in this case is considerably displaced from the anti-symmetric stretch orbit, the trajectory must necessarily couple to the symmetric stretch mode. If we were to launch the wavepacket at the outer slope of the saddle point, the anti-symmetric stretch periodic orbit would support recurrences by itself without coupling to the symmetric stretch mode. An example is the dissociation of IHI discussed in Section 7.6.2. 

All the peaks in the real part of the i -transform can be assigned to classical periodic orbits (see labels in Fig. 10.10). As predicted by (10.4.31) the first peak in Fig. 10.10 occurs at half the action of the 1 1 periodic orbit. A peak corresponding to the symmetric stretch orbit is missing. This was first noticed by Ezra et al. (1991). 

The symmetric stretch orbit is not the only periodic orbit emerging from the point x = = 0. There is a whole family of other periodic... 

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

In the presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A = 0, +, - [10]. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. 

The derivation of (4.13) shows that the equilibrium point quantization and the periodic-orbit quantization can be compared term by term. This comparison shows that the periodic-orbit quantization is able to take into account the anharmonicities in the direction of symmetric stretch. However, the anhar-monicities are neglected in the other directions transverse to the periodic orbit. Their full treatment requires the calculation of h corrections to the Gutzwiller trace formula, as shown elsewhere [14]. 

Figure 15. Three-branch Smale horseshoe in the 2F collinear model of Hgl2 dissociation at the energy E = 600 cm 1 above the saddle in a planar Poincare surface of section transverse to the symmetric-stretch periodic orbit. The Smale horseshoe is here traced out in a density plot of the cumulated escape-time function (4.6).

Figure 16. Scattering resonances of the full rotational-vibrational Hamiltonian describing the dissociation of CO2 on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with 7 = 0,..., 10 are given by dots. Their close vicinity explains the formation of hyphens , i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with V2 = 0,. .. 5. The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

In a more recent work, Joens [158] has assigned the structures of the Hartley band using a Dunham expansion, that is equilibrium point quantization. The lifetime predicted by his analysis is extremely short, equal to 3.2 fs, while the symmetric stretching period is of 30 fs. Recall, however, that the interpretations in terms of equilibrium point expansions and in terms of periodic orbits are strictly complementary only for regular regimes. 

To answer Prof. Marcus s question, we may therefore conclude that the natural motions of the system are the short-time periodic orbits. Those that arise from the symmetric-stretch bifurcations depend on the frequency ratio local modes in the 1 1 case, 7-shaped orbits at the 3 2 instability, horseshoes at the 2 1 resonances, and so on. 

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