Quasiperiodic motion

Hose G and Taylor H A 1982 A quantum analog to the classical quasiperiodic motion J. Chem. Phys. 76 5356-64... 

In the present volume we discuss only two limits of coherent and incoherent transitions, and have not considered the intermediate region in which the coherence is destroyed. In the classical dynamics of polyatomics, the periodicity of motion is destroyed due to increasing coupling with the bath modes, which leads to quasiperiodic motion and... 

The power sjjectrum is merely the Fourier transform of the VAF via the Wiener-Khintchine theorem. The integration is carried out as a discrete sum over the jjeriod of time in which the VAF decays to a zero value. This quantity gives the number of oscillators at a given frequency and is a very informative indicator of the transition from rigid, quasiperiodic motion to nonrigid, chaotic motion. Note that I(co = 0) is proportional to the diffusion constant. This quantity was calculated by Dickey and Paskin - in the study of phonon frequencies in solids and also by Kristensen et al. in simulations of cluster melting. 

It is clear, from the discussion thus far, that typical molecular Hamiltonians display features characteristic of chaotic classical motion. The logical order followed previously, that is, the introduction of well-defined concepts of chaotic behavior in classical ideal systems, followed by an examination of realistic molecular models, does not follow through to quantum mechanics. The primary difficulty is that quantum mechanics always predicts, for bound-state dynamics, quasiperiodic motion. Several aspects of quantum chaos are discussed in Section IV. We note at this point, however, that this quantum-... 

The correspondence between classical and quantum mechanics tells us that this trajectory corresponds to a resonance state with a localized wave function in the H-C-C vibration continuum. The quantum mechanical resonance state (discussed in detail in Section 15.2.4) will have nearly the same energy and be assignable with the semiclassical quantum numbers. It is also expected to have a very long lifetime as a result of the classical quasiperiodic motion. 

For some systems quasiperiodic (or nearly quasiperiodic) motion exists above the unimolecular threshold, and intrinsic non-RRKM lifetime distributions result. This type of behaviour has been found for Hamiltonians with low unimolecular thresholds, widely separated frequencies and/or disparate masses [12,62,65]. Thus, classical trajectory simulations performed for realistic Hamiltonians predict that, for some molecules, the unimolecular rate constant may be strongly sensitive to the modes excited in the molecule, in agreement with the Slater theory. This property is called mode specificity and is discussed in the next section. 

Solving the classical equations of motion, Eq. (2.9), for this Hamiltonian gives rise to quasiperiodic motion in which each normal mode coordinate Qj varies with time according to... 

Figure 11 Surface of section plots for the De Leon—Berne Hamiltonian as a function of energy for energies below the barrier. Note the destruction of quasiperiodic motions (KAM tori) as a function of energy. Also note the persistence of ICAM tori near certain elliptic fixed points. Reprinted with permission from Ref. 119.

The study of the signatures of classical chaos in the quantum mechanical description of a general system is too complex for us to undertake at present. However, the phase space structure of a classical system that is exclusively defocussing is simpler than that of a general system. In particular, in an exclusively defocussing system the quasiperiodic motions of type (i) are absent. Examples of exclusively defocussing systems are the elastic collisions of a point particle with an assembly of hard discs or hard spheres or, indeed, any hard objects with smooth convex boundaries. 

Quasiperiodic trajectories are a special case of Poisson-stable trajectories. The latter plays one of the leading roles in the theory of dynamical systems as they form a large class of center motions in the sense of Birkhoff (Sec. 7.2). Birkhoff had partitioned the Poisson-stable trajectories into a number of subclasses. This classification is schematically presented in Sec. 7.3. Having chosen this scheme as his base, as early as in the thirties, Andronov had undertaken an attempt to collect and correlate all known types of dynamical motions with those observable from physical experiments. Since his arguments were based on the notion of stability in the sense of Lyapunov for an individual trajectory, Andronov had soon come to the conclusion that all possible Lyapunov-stable trajectories are exhausted by equilibrium states, periodic orbits and almost-periodic trajectories (these are quasiperiodic and limit-quasiperiodic motions in the finite-dimensional case). 

Thus, it appeared naturally to assume that every interesting dynamical regime possesses a discrete frequency spectrum. In this connection, it is curious to note that Landau and Hopf had proposed quasiperiodic motions with a sufficiently large number of independent frequencies as the mathematical image of hydrodynamical turbulence (the number of the frequencies was supposed to increase to infinity as some structural parameter, such as the Reynolds number, increases). 

Therefore, Andronov s approach (Sec. 8.4) for studying dynamical models has to be corrected in cases where a complete bifurcation analysis may not be possible without moduli. We note, however, that if some fine delicate phenomena may be ignored, or if the problem is restricted to the analysis of non-wandering orbits like equilibrium states, periodic and quasiperiodic motions, a study of the main bifurcations in systems with simple dynamics still remains realistic within the framework of finite-parameter families under certain reasonable requirements (Sec. 8.4). 

As for attractive minimal sets, it follows from Pugh s theorem that they are structurally unstable. Although the minimal sets composed of recurrent and limit-quasiperiodic orbits are by far not key players in the nonlinear dynamics, quasiperiodic motions have always been of major interest because they model many oscillating phenomena having a discrete spectrum. 

C-H stretch) can be used. Thus, it appears that both stochastic and quasiperiodic motion can be adequately treated by numerical integrations with large step sizes. 

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