Phase space quasiperiodic

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

Thus some of the fluid elements move on aperiodic chaotic trajectories and others on quasiperiodic orbits. The quasiperiodic orbits are invariant surfaces in the phase space that form the boundaries of the chaotic layers and limit the motion of the chaotic trajectories. There is a similar structure around each elliptic periodic orbit resulting from broken resonant tori that are also surrounded by invariant tori forming isolated islands inside the chaotic region. 

In general, a Hamiltonian is neither completely integrable, nore purely chaotic, which means that, in practice, phase space fragments itself into islands of stability, inside which the motion is quasiperiodic, and regions,... 

Although the phase space trajectories appear as simple curves on the two-dimensional Iz,ip phase space diagram (the 0 coordinate is suppressed) most trajectories are actually quasiperiodic. The actual trajectories he on the 2-dimensional surface of a 3-dimensional invariant torus in 4-dimensional phase space. Fig. 9.14 shows such a torus. Any point on the surface of the torus is specified by two angles, 0 and. The 0 and circuits about the torus are shown, respectively, as large and small diameter circles. The diameter of the 0... 

The fixed points on the phase space diagrams or phase spheres in Fig. 9.13 are labeled A, B, Ca, and C. Each corresponds to a periodic orbit that is said to organize the surrounding region of phase space that is filled with topologically similar quasiperiodic trajectories. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

Analysis of this 7feff using the techniques of nonlinear classical dynamics reveals the structure of phase space (mapped as a continuous function of the conserved quantities E, Ka, and Kb) and the qualitative nature of the classical trajectory that corresponds to every eigenstate in every polyad. This analysis reveals qualitative changes, or bifurcations, in the dynamics, the onset of classical chaos, and the fraction of phase space associated with each qualitatively distinct class of regular (quasiperiodic) and chaotic trajectories. 

The structure of phase space accessible at specified values of Ka, Kb, and E is revealed on the surface of section by launching a series of trajectories and, for each trajectory, plotting a point on the (Jb, ipb) plane at each location where the trajectory crosses the plane in the > 0 direction. If the trajectory is quasiperiodic, the set of all of its intersections with the (Jb,ipb) plane forms a closed curve on the surface of section. 

Each trajectory is launched at chosen initial values of Jb and ipb and at fra = 0. Since any point on the 3-dimensional energy shell may be specified by three linearly independent coordinates, selection of initial values J , ip%, and ip°, implies a definite value of J°. Thus trajectories are launched at various [Jfi, ip%, ip° = 0, J°(J , ipl, ip°,) ] initial values until all of the qualitatively distinct regions of phase space are represented on the surface of section by either a family of closed curves (quasiperiodic trajectories) that surround a fixed point (a periodic trajectory that defines the qualitative topological nature of the neighboring quasiperiodic trajectories) or an apparently random group of points (chaos). Often, color is used to distinguish points on the surface of section that belong to different trajectories. 

At low levels of excitation, the motion associated with molecular Hamiltonians is quasiperiodic. However, as the energy is increased, in most cases there is a gradual destruction of the tori and a transition from quasiperiodic to chaotic motion, with both types present at intermediate energies. If all the tori are destroyed, all regions of phase space become accessible to a trajectory, and the ergodic hypothesis becomes valid. 

As discussed above (section 4.3.2) these local-mode states are not the eigenstates for the system, but superposition states. However, since the classical motion is quasiperiodic for these local mode states (i.e., the state is a torus in the phase space), the system is trapped in the initially excited local-mode state and the quantum periodic oscillation between the n,m) and (m,n) local mode states is not observed classically. Thus, classical mechanics severely underestimates the rate of energy transfer. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

A great deal of attention has been focused in recent years by workers in classical dynamics on the geometric properties of phase space structures and their manifestation on Poincare maps (also referred to as surfaces of section). The result has been the blossoming of a huge literature on the subject of nonlinear dynamics (quasiperiodicity and dynamical chaos), which is discussed in a number of recent textbooks and articles. - ... 

In Figure 10 the chaotic region is extremely small. However, in Figures 11 and 12 we show a second system s phase space map as a function of en-ergy.35,119 xhjs system exhibits a mode-mode resonance at low energies, with a hyperbolic fixed point located near the center of the Poincare map. Note in Figure 11 that as the energy increases, the measure of quasiperiodic phase space decreases and approaches a limit in which most of the tori are destroyed, with... 

FIGURE 11 Sample quasiperiodic trajectory in a two-degrees-of-freedom system as it moves on the surface of a torus in phase space. The trajectory shown is actually periodic in general, the trajectory will fill the entire torus surface. 

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. 

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