Liapunov exponent

Lewis number, 27 62, 64, 66 Liapunov exponent, 39 115 Lifetime effects, 34 213 Ligand... 

Fig. 16. A chaotic front moving through a fixed-bed reactor during CO oxidation over Pt/ AljOj. Ta, Tk, Tc, and Tj are pellet temperatures in the entrance cross section T) and T2 are thermocouple temperatures 24 and 46 mm downstream, respectively. Chaos was characterized by correlation dimension and Liapunov exponent. (From Ref. 700.)...

Fig. 17. The transition to chaos (from a to d) observed in the work function during CO oxidation on Pt(l 10) while decreasing CO pressure. Chaos in the upper time series (d) was characterized by the Liapunov exponent, Kolmogorov entropy, and the embedding dimension (From Ref. 68.)...

Fig. 20. Largest Liapunov exponent versus heat transfer coefficient for gas-phase coupling K for the NO/CO reaction modeled on a catalyst wafer. The wafer is composed of 100 cells with 10 randomly distributed active cells as shown on the grid. The numbers pointing at various regions indicate the onset of particular periodicities chaos is observed for 0.15 < i < 18. (From Ref. 232.)...

The three-disc system is an example of a hyperbolic scattering system. For a scattering system to be hyperbolic it must contain an uncountable set of trapped trajectories which are all unstable, i.e. the Liapunov exponent describing the rate of exponential growth of infinitesimal deviations from a given trajectory is positive for each trapped trajectory. Hyperbolicity also requires that the trapped trajectories are uniformly unstable, meaning that there is a finite positive lower bound to all Liapunov exponents [39]. These conditions are fulfilled for the three-disc system [29]. 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

Fig. 12, Liapunov exponents of the unstable periodic orbits labelled — + + +, where each period contains one slow Coulombic oscillation of the farther electron and n Coulombic oscillations of the closer electron in between two crossings of the line ri = T2 in coordinate space. The ordinate measures the products n2 on a linear scale and the abscissa measures n on a logarithmic scale. The solid dots show the results for s-wave helium (Eq. (16)) [55], the open circles show the corresponding results for collinear helium (Eq. (15)) [52]. The linear behaviour for large n illustrates the proportionality of /l to (log n)/n. (Results are for charge Z = 2 in the Hamiltonian, Eq. (14))...

The stability exponents labeled u in Table 1 of Wintgen et al. [52] are actually the Liapunov exponents of the periodic orbits multiplied by their respective periods, u = 2T.]... 

III. Power Spectra, Phase-Space Dimensions, Liapunov Exponents, and Kolmogorov Entropy... 

We may look at atomic clusters as particularly apt and useful models to study virtually every aspect of what we call, diffusely, complexity. Simulations are particularly powerful means to carry out such studies. For example, we can follow trajectories for very long times with molecular dynamics and thereby evaluate the global means of the exponential rates of divergence of neighboring trajectories. This is the most common way to evaluate those exponents, the Liapunov exponents. The sum of these is the Kolmogorov entropy, one gross measure of the volume of phase space that the system explores, and hence one... 

III. POWER SPECTRA, PHASE-SPACE DIMENSIONS, LIAPUNOV EXPONENTS, AND KOLMOGOROV ENTROPY... 

Next, we turn to the Liapunov exponents and the Kolmogorov entropy (K-entropy). The systems concerning us here are Hamiltonian, so the Liapunov... 

Figure 7. Two distributions of sample Liapunov exponents for a cluster of three Lennard-Jones particles simulating Ar3 at energy equivalent to 4.15 K. The only difference between the two calculations is in the initial conditions. [Reprinted with permission from C. Amitrano, and R. S. Berry, Phys. Rev. Lett. 68, 729 (1992). Copyright 1992, American Physical Society.]...

A geometry that can actually materialize such a memory-losing dynamics is studied in Section IV. We here propose a notion of inter-basin mixing that is responsible for the Markov-type stochastic appearance of molecular structures in the above memory-losing isomerization dynamics. An extension of the Liapunov exponent to quantify the time scale to reach inter-basin mixing is also proposed [10]. 

In a similar context, Nayak and Ramaswamy have shown that the maximum Liapunov exponent (MLE) rises very steeply just as the Lindemann index and thereby can detect the aforementioned transition very well [20]. Since MLE is well established to measure the exponential divergence of the distance between nearby trajectories in phase space [21,22], their numerical results seem to suggest that the phase-transition could be a consequence of strong chaos behind the dynamics. We henceforth examine the resultant phenomena, the geometry on which the dynamics is characterized, and a statistical law of associated isomerization reaction. 

The constant (3 can be regarded as a Liapunov exponent, the inverse of which gives a time scale for a system to reach the state of inter-basin mixing. In Fig. 8, we have p 0.17 or ( 1 6.0. And, at time t = 6, it is observed in Fig. 9 that Gnax(6) = 0.05. This means rmax(6) = 0.37 x rmax(0), but since the dimensionality of the cross section of a phase space is 29, the area of the cross section of this reaction tube is already as small as roughly rmax(6)29 3 x 10-13 x rmax(0)29. 

The configuration entropy represents the size of the phase space, and its projection onto the configuration space may correspond to the extent of the configuration space. However, it does not give any information about the dynamics itself. The Liapunov exponents and the KS entropy that is the positive sum of the Liapunov exponents may give the characteristic of the dynamics. Thus, these properties were calculated in this work to compare the results... 

Figure 14. (a) The KS entropy that is the positive sum of the Liapunov exponents against the... 

The number A is often called the Liapunov exponent, although this is a... 

First, there are actually n different Liapunov exponents for an n-dimensional system, defined as follows. Consider the evolution of an infinitesimal sphere of perturbed initial conditions. During its evolution, the sphere will become distorted into an infinitesimal ellipsoid. Let 5j(r), k = l,...,zz, denote the length of the th principal axis of the ellipsoid. Then 5j(r) 5 (0), where the Aj are the Liapunov exponents. For large t, the diameter of the ellipsoid is controlled by the most positive Aj. Thus our A is actually the largest Liapunov exponent. 

When a system has a positive Liapunov exponent, there is a time horizon beyond which prediction breaks down, as shown schematically in Figure 9.3.6. (See Lighthill 1986 fora nice discussion.) Suppose we measure the initial conditions of an experimental system very accurately. Of course, no measurement is perfect— there is always some error <5fl between our estimate and the true initial state. 

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