Chaotic trajectories

The reduction of chaos for 9 = 1.45 is presented in the intensity portraits of Fig. 39. However, as is seen in Fig. 38a, there is a small region (1.68 < 9 < 1.80) where the system behaves orderly in the classical case but the quantum correction leads to chaos. By way of an example for 9=1.75, the classical system, after quantum correction, loses its orderly features and the limit cycle settles into a chaotic trajectory. Generally, Lyapunov analysis shows that the transition from classical chaos to quantum order is very common. For example, this kind of transition appears for 9 = 3.5 where chaos is reduced to periodic motion on a limit cycle. Therefore a global reduction of chaos can be said to take place in the whole region of the parameter 0 < 9 < 7. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

At interesting phenomenon occurs in the case of other resonance horns we have studied it for the case of the 3/1 resonance. The torus pattern breaks when the subharmonic periodic trajectories locked on it for small FA decollate from the torus as FA increases. We are left then with two attractors a stable period 3 and a stable quasi-periodic trajectory. This is a spectacular case of multistability (co-existence of periodic and quasi-periodic oscillations). The initial conditions will determine the attractor to which the system will eventually converge. This decollation of the subharmonics from the torus was predicted by Greenspan and Holmes (1984). They also predicted chaotic trajectories close to the parameter values where the subharmonic decollation occurs. 

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. 

Figure 3. Intersection of chaotic trajectories with plane Br = 0.

A Poincare surface of section may be used to identify the chaotic and quasi-periodic regions of phase space for a two-dimensional Hamiltonian. An ensemble of trajectories, chosen to randomly sample the phase space, are calculated and for each trajectory a point is plotted in the (9i,Pi)-plane every time Q2 = 0 for p2 > 0. A quasi-periodic trajectory lies on an invariant curve, while the points are scattered for a chaotic trajectory with no pattern. Figure 44 shows an example for a two-dimensional model for HOCl the HO bond distance is frozen in these calculations [351]. It clearly illustrates how the phase space becomes gradually more chaotic as the energy increases. 

For quasi-periodic trajectories, like those for the normal-mode Hamiltonian in Eq. (69), I to) consists of a series of lines at the frequencies for the normal modes of vibration. In contrast, a Fourier analysis of a chaotic trajectory results in a multitude of peaks, without identifiable frequencies for particular modes. An inconvenience in this approach is that for a large molecule with many modes, a trajectory may have to be integrated for a long time T to resolve the individual lines in a power spectrum for a quasi-periodic trajectory. Moreover, in the presence of a resonance between different modes, the interpretation of the power spectrum may become misleading. 

The nature of the intramolecular motion may also be identified by studying the way the separation of two trajectories evolves with time [353]. If the motion is regular (quasi-periodic) the separation is linear with time, but exponential if the motion is irregular (chaotic). If the separation is exponential, the rate of the separation — called the Lyapunov characteristic exponent — provides qualitative information concerning the IVR rate for the chaotic trajectories. This type of analysis has been reported, for example, for NO2 [271] and the Cl CHsBr complex [354]. 

The simplest approach [338] to describe a non-ergodic unimolecular system is to assume that the reactant s phase space only consists of quasi-periodic and chaotic trajectories, whose numbers are ATqp and Nch- If a micro-canonical ensemble is prepared at t = 0 and if it is assumed that a restricted micro-canonical ensemble is maintained within the chaotic region, while no trajectory dissociates from the quasi-periodic region, the number of reactant molecules versus time is... 

Figure 8. Center-of-mass trajectories obtained from a simulation of a patch of nitrogen molecules adsorbed on the basal plane of graphite. (Carbon atoms are shown by the small dots.) In part (a) for T=36.9 K, the molecules are commensurate with theVSxVS lattice and vibrate around the site centers except at the edges of the patch. In part (b), 7M4.0 K and the patch has mehed to a 2D fluid that is characterized by chaotic trajectories in the him. (At longer simulation times, the molecules appear to fill the surfece as a 2D gas.) From Ref. [38], Mol. Phys. 55 (1985) 999-1016.

Roux et al. (1983) also considered the attractor in three dimensions, by defining the three-dimensional vector x(r) = (B(r),B(r+T),B(r+2T)). To obtain a Poincare section of the attractor, they computed the intersections of the orbits x(r) with a fixed plane approximately normal to the orbits (shown in projection as a dashed line in Figure 12.4.2). Within the experimental resolution, the data fall on a one-dimensional curve. Hence the chaotic trajectories are confined to an approximately two-dimensional sheet. 

Figure 6. Hypercube structure of a four-dimensional network with a chaotic attractor. One of the two cycles followed by the chaotic trajectory is marked by bold lines.

For each set of initial conditions, Eqs. (4.1)-(4.3) can be solved to And X ", U ", and The initial conditions are randomly selected from known distribution functions, and we can assume that there is an infinite number of possible combinations. Each combination is called a realization of the granular flow, and the set of all possible realizations forms an ensemble. Note that, because the particles have finite size, they cannot be located at the same point thus X " 4 X for n 4 m. Also, the collision operator will generate chaotic trajectories and thus the particle positions will become uncorrelated after a relatively small number of collisions. In contrast, for particles suspended in a fluid the collisions are suppressed and correlations can be long-lived and of long range. We will make these concepts more precise when we introduce fluid-particle systems later. While the exact nature of the particle correlations is not a factor in the definition of the multi-particle joint PDF introduced below, it is important to keep in mind that they will have... 

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. 

Thus, the time-dependence of the flow generates chaotic trajectories that will enhance the mixing of fluid within these regions. However, the KAM tori formed by the remaining quasiperiodic orbits separate the domain into a set of disconnected regions with no advec-tive transport between them. Therefore, when the time-dependence is weak the fluid is only mixed within narrow layers around the resonant streamlines of the original time-independent flow. The areas... 

Since the time-dependence of the velocity field is restricted to a finite region the complex chaotic orbits are also limited to this region. Advection in such flows is a chaotic scattering process (Ott and Tel, 1993 Ott, 1993) in which fluid elements approach the mixing zone along the inflow streamlines, they follow chaotic trajectories inside... 

Note that the chaotic scattering can only take place when the typical lifetime of the transient chaotic advection, 1/k, is longer than the rate of separation of chaotic trajectories on the chaotic saddle, 1/A. Otherwise the particles escape before they could be shuffled within the mixing zone. Thus k < A is a necessary condition for the chaotic scattering with an exponential escape time distribution. From this it follows that the dimensions of the manifolds in two-dimensional flows satisfy 1 < D s < 2. 

Neufeld et al. (1999) have shown that the roughness exponent a of the decaying chemical field is a function of the decay rate and of the Lyapunov exponent of the advection. In the large Peclet number limit we can neglect diffusion and set D = 0 so that the concentration field can be described by the Lagrangian representation (6.13) that follows the chemical dynamics within the fluid parcels advected on chaotic trajectories in the flow... 

Fig. 10.1. Showing (a) a closed orbit (b) multiperiodic orbits on an invariant torus and (c) chaotic trajectories (after D. Pfenniger [527]).

These calculations reveal another interesting property. As the complexity of the plots increases, chaotic trajectories appear, which grow (as expected) from the separatrix outwards. The presence of such orbits heralds new paths for ionisation, which may eventually dominate the whole of phase space. 

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. 

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