Recurrent trajectory

The Poisson-stable trajectories may be sub-divided into two kinds depending on whether the sequence Tfc(e) of Poincare return times of a P-trajectory to its -neighborhood is bounded or not. Birkhoff named the trajectories of the first kind recurrent trajectories. Such a trajectory is remarkable because regardless of the choice of the initial point, given e > 0 the whole trajectory lies in an -neighborhood of the segment of the trajectory corresponding to a time interval L(e). Obviously, equilibrium states and periodic orbits are the closed recurrent trajectories. 

The relationship between a minimal set and a recurrent trajectory is constituted by the following theorems. 

Theorem 7.4. (Birkhoff) The closure of a recurrent trajectory is a minimal set. 

In the case of recurrent trajectories, there are certain statistics in Poincare return times which are weaker than that characterizing genuine Poisson-stable trajectories. Nevertheless, there is a particular sub-class of recurrent trajectories which is interesting in nonlinear dynamics. This is the class of the so-called almost-periodic motions. The remarkable feature which reveals the origin of these trajectories is that each component of an almost-periodic motion is an almost-periodic function (whose analytical properties are well studied, see for example [49, 66, 84]). 

Theorem 7.5. (Franklin, Markov) If a recurrent trajectory possesses the S-property, then it is almost-periodic. 

One of the conclusions from this theorem is that an authentic recurrent trajectory must be unstable. A few exotic examples of dynamical systems on some compact manifolds, called nil-manifolds, are known where all trajectories are recurrent. Moreover, these trajectories are unstable. However, their instability is not exponential but only polynomial. In contrast to an almost-periodic trajectory whose frequency spectrum is discrete, the spectrum of a recurrent trajectory has in addition a continuous component. For further details see [23]. 

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

The origin of this resonance was identified by extending Reinhardt s21 wave packet notion. Realizing that the wave packet evolves along the classical trajectories of the electron, Holle et al. searched for classical trajectories in which the electron leaving the origin returned in a time of 9.5 ps, 1/2tt (0.64[Pg.153]

Another recent trend is to show the importance of hydrophobic profiles rather than molecular hydrophobicity. Giuliani et al. (2002) suggested nonlinear signal analysis methods in the elucidation of protein sequence-structure relationships. The major algorithm used for analyzing hydrophobicity sequences or profiles was recurrence quantification analysis (RQA), in which a recurrence plot depicted a single trajectory as a two-dimensional representation of experimental time-series data. Examples of the global properties used in this... 

In the time-dependent picture, resonances show up as repeated recurrences of the evolving wavepacket. Resonances and recurrences reveal, in different ways, the same dynamical effect, namely the temporary excitation of internal motion within the complex. In the context of classical mechanics, the existence of quantum mechanical resonances is synonymous with trapped trajectories performing complicated Lissajou-type motion before they finally dissociate. The larger the lifetime, the more frequently the wavepacket recurs to its starting position, and the narrower are the resonances. 

The transition from direct to indirect photodissociation proceeds continuously (see Figure 7.21) and therefore there are examples which simultaneously show characteristics of direct as well as indirect processes the main part of the wavepacket (or the majority of trajectories, if we think in terms of classical mechanics) dissociates rapidly while only a minor portion returns to its origin. The autocorrelation function exhibits the main peak at t = 0 and, in addition, one or two recurrences with comparatively small amplitudes. The corresponding absorption spectrum consists of a broad background with superimposed undulations, so-called diffuse structures. The broad background indicates direct dissociation whereas the structures reflect some kind of short-time trapping. 

According to Section 4.1.1 the wavepacket is a superposition of stationary wavefunctions corresponding to a relatively wide range of energies. This and the superposition of three apparently different types of internal vibrations additionally obscures details of the underlying molecular motion that causes the recurrences. A particularly clear picture emerges, however, if we analyze the fragmentation dynamics in terms of classical trajectories. 

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. 

The quantum mechanical wavepacket closely follows the main classical route. It slides down the steep slope, traverses the well region, and travels toward infinity. A small portion of the wavepacket, however, stays behind and gives rise to a small-amplitude recurrence after about 40-50 fs. Fourier transformation of the autocorrelation function yields a broad background, which represents the direct part of the dissociation, and the superimposed undulations, which are ultimately caused by the temporarily trapped trajectories (Weide, Kiihl, and Schinke 1989). A purely classical description describes the background very well (see Figure 5.4), but naturally fails to reproduce the undulations, which have an inherently quantum mechanical origin. 

Fig. 8.11. (cont.) the unstable periodic orbit, represented by the solid line, influences the dissociation dynamics all direct trajectories, which fragment immediately without any recurrence, are discarded. The times range from 0 fs in (a) to 50.8 fs in (h). The arrows schematically indicate the evolution of the classical wavepacket and the heavy dot marks the equilibrium of the R-state potential energy surface. Adapted from Weide, Kiihl, and Schinke (1989). 

This is inherently impossible in the time-independent approach because the wavefunction contains the entire history of the wavepacket. The real understanding, however, is provided by classical mechanics. Plotting individual trajectories easily shows the type of internal motion leading to the recurrences which subsequently cause the diffuse structures in the energy domain. The next obvious step, finding the underlying periodic orbits, is rather straightforward. 

The previous discussion (Ma, 1985) considered a lattice (Ising) model as a physical example and focused the concepts on recurrences in the trajectory. A method of evaluating the entropy was suggested. Here we carry that suggestion further, and discuss how that idea can be used to estimate values of integrals. We consider a classic quadrature problem, evaluation of an integral such as... 

Polva s Recurrence Theorem dictates one of the most important differences between 7.B and AB. It states that every random-walk trajectory in one or two dimensions passes through every point in space but that this is not true in three dimensions 17 J.SO. The Recurrence Theorem is the subject of a famous (among mathematicians) joke. 

Classical trajectories may either under- or overestimate the rate of IVR. Some relaxation processes are not allowed classically for example, quantum mechanical tunneling through potential energy barriers. Also, in the absence of such a barrier classical mechanics may still not allow an initial zero-order state to relax, even though the state is quantum-mechanically nonstationary. In the other extreme classical mechanics may be more chaotic than quantum dynamics. Quantum mechanics often gives more structured motion with more recurrences among zero-order states than does classical mechanics. Each of these extremes is illustrated in the following. 

From a time-dependent point of view, recurrences in the probability of occupying the initially prepared state give rise to the fine structure in the overtone absorption spectrum. Though rudiments of these recurrences may be present in the short-time trajectory P n,t), chaotic classical motion destroys the longer time recurrences, which occur quantum mechanically. It is these latter recurrences which are needed to evaluate fine details in the absorption spectrum. Thus, the classical trajectory method may be limited to the evaluation of low-resolution absorption spectra. However, it should be pointed out that progress is being made in extracting information from systems with... 

The time evolution of the atoms in a cluster is simulated by their classical trajectories using the Verlet algorithm (cf. Ref. [25]). According to this algorithm, the position and velocity of the nucleus /(r/, Vj) at time step tn = nAt, are obtained recurrently ... 

The central sub-class of non-wandering points are points which are stable in the sense of Poisson. The main feature of a Poisson-stable point is not only the recurrence of its neighborhood but the recurrence of the trajectory itself. The definition of Poisson-stable points below is different in some ways but equivalent to the definition given in Chap. 1. 

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