Unstable periodic orbits

Proposition 1. For e sufficiently small the periodic orbit 7(t) of (7) survives as aperiodic orbit 7 (t) = rj(t)+0(e), of (10) having the same stability type as 7(t), and depending on e in a C2 manner. Moreover, the local stable and unstable manifolds Wlsoc( ye(t)) and Wfoc e(t)) of 7 (t) remain also e-close to the local stable and unstable manifolds WfoMt)) and Wfioc( (t)) of ft), respectively. 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

Another well-defined configuration of the classical three body Coulomb problem with unambiguous quantum correspondence is the collinear antisymmetric stretch configuration, where the electrons are located on opposite sides of the nucleus. In contrast to the frozen planet orbit, the antisymmetric stretch is unstable in the axial direction (G.S. Ezra et.al., 1991 P. Schlagheck et.al., 2003), with the two electrons colliding with the nucleus in a perfectly alternating way (Fig. 3 (left)). Hence, already the one dimensional treatment accounts for the dominant classical decay channel of this configuration. As for the frozen planet, there are doubly excited states of helium associated to the periodic orbit of the ASC as illustrated in Fig. 3 (left). 

Although the phase space of the nonadiabatic photoisomerization system is largely irregular, Fig. 36A demonstrates that the time evolution of a long trajectory can be characterized by a sequence of a few types of quasi-periodic orbits. The term quasi-periodic refers here to orbits that are close to an unstable periodic orbit and are, over a certain timescale, exactly periodic in the slow torsional mode and approximately periodic in the high-frequency vibrational and electronic degrees of freedom. In Fig. 36B, these orbits are schematically drawn as lines in the adiabatic potential-energy curves Wo and Wi. The first class of quasi-periodic orbits we wish to consider are orbits that predominantly... 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

The periodic-orbit contribution derived by Gutzwiller is general and applies to different kinds of periodic orbits. However, the applicability of (2.13) rests on the property that the periodic orbits are isolated, that is, they do not belong to a continuous family. This is the case in hyperbolic dynamical systems where all the periodic orbits are linearly unstable. We should emphasize that the Gutzwiller trace formula may apply both to bounded and scattering systems. 

The terms that occur high in the series have a small amplitude and contribute little. Therefore, the zeros of the zeta function will be determined essentially by the first few terms associated with the least unstable periodic orbits. Contrary to systems with one degree of freedom, no factorization is possible so that the different periodic orbits have additive contributions that interfere. The distribution of zeros will therefore have the tendency to become irregular, contrary to classically integrable systems. 

The concentration of the Wigner transforms of the eigenfunctions on the quantized tori provides the counterpart in classically integrable systems to the scarring phenomena associated with unstable periodic orbits. 

Due to the nonlinearities of the classical Hamiltonian, the periodic orbits undergo bifurcations at critical energies. At these bifurcations, the stability of the orbit changes and extra periodic orbits are created or existing ones annihilated [19]. These bifurcations have dramatic effects on the semiclassical amplitudes of the periodic orbits [49]. In particular, the comparison between the amplitudes of neutrally stable and unstable periodic orbits shows that the amplitude is expected to be globally lowered after a destabilization. 

Moreover, the Poincare mappings of (3.14) at values of Pc) fixed by the existence of (5, 6) and I4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E - 6900 cm-1 (see Fig. 6). At this bifurcation, the periodic orbit (5, 6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6, 7) destabilizes by a similar scenario around E - 7200 cm-1. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-... 

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. 

Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwarder backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller [19, 33, 35, 48]. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Reciprocally, the trajectories that approach the... 

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal iepeller. As we mentioned in Section II, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodic-orbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. 

Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions, however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character. 

---