Quadratic map

As we have already commented, mappings of the type discussed above are not in any way easily related to a given set of reaction rate equations. Such mappings have, however, been used for chemical systems in a slightly different way. A quadratic map has been used to help interpret the oscillatory behaviour observed in the Belousov-Zhabotinskii reaction in a CSTR. There, the variable x is not a concentration but the amplitude of a given oscillation. Thus the map correlates the amplitude of one peak in terms of the amplitude of the previous excursion. 

At this point 1 was reminded by Paul Stein that period-doubling isn t a unique property of the quadratic map but also occurs, for example, in... 

The method we have described is simple and effective, but we do not claim it is the final word on automated computer art. You can explore other criteria for selecting the patterns and other ways to display them. We chose a generalized two-dimensional quadratic map to emphasize the complexity and variety that arises from simple equations. You can easily extend the technique to other mathematical functions and to higher dimensions. The same ideas can be used for automatic generation and evaluation of computer music. We hope that we have introduced you to a new use for your computer, and we would like to hear of any interesting results you obtain. 

Marrero-Ponce, Y. lyarreta-Veitia, M. Montero-Torres, A. Romero-Zaldivar, C. Brandt, C. A. Avila, P. E. Kirchgatter, K. Machado, Y. Ligand-based virtual screening and in silico design of new antima-larial compounds using nonstochastic and stochastic total and atom-type quadratic maps. J. Chem. Inf. Model. 2005,45,1082-1100. 

As the quadratic map teaches us, before the onset of chaos a sequence of period doublings occurs.This has become one famous route to chaos and has been found in quite different kinds of systems such as in fluids, lasers, and electronic devices [21]. Another way is called the Ruel1e-Takens route in which a system first undergoes an oscillation at a frequency cdi a further oscillation at a second frequency o)2 occurs in addition, and afterwards chaos should set in in the "generic" case. I have pointed out for a number of years [23] that the question of whether chaos occurs or whether still more incommensurate frequencies in a quasi periodic motion occur, is not a question of "genericity", but is a quantitative... 

This map has a single quadratic extremum, similar to tliat of tire WR model described in detail earlier. Such maps (togetlier witli tire technical constraint of negative Schwarzian derivative) [23] possess universal properties. In particular, tire universal (U) sequence in which tire periodic orbits appear [24] was observed in tire BZ reaction in accord witli tliis picture of tire chemical dynamics. 

Figure 5.2 shows the finite element mesh corresponding to the configuration shown in Figure 5.1. This mesh consists of 225 nine-node bi-quadratic elements and its utihzation in the present model is based on the application of isoparametric mapping, described in Chapter 2.

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

Figure 34. Contour map of the nonadiabatic transition probability Pn induced by quadratically chirped pulse as a function of the two basic parameters a and p. Taken from Ref. [37]-...

The primary difficulty in using the SOM, which we will return to in the next chapter, is the computational demand made by training. The time required for the network to learn suitable weights increases linearly with both the size of the dataset and the length of the weights vector, and quadratically with the dimension of a square map. Every part of every sample in the database must be compared with the corresponding weight at every network node, and this process must be repeated many times, usually for at least several thousand cycles. This is an incentive to minimize the number of nodes, but as the number of nodes needed to properly represent a dataset is usually unknown, trials may be needed to determine it, which requires multiple maps to be prepared with a consequent increase in computer time. 

Figure 2. The histogram is the distribution of OH stretch frequencies for the water clusters and surrounding point charges, and the solid line is the distribution of frequencies from the quadratic electric field map.

IR and Raman line shapes have been measured for H0D/H20. They peak near 2500 cm 1 and have line widths in the 160 to 180 cm 1 range. Corcelli et al. [151] calculated these line shapes using the approaches described in Section III.C, for the SPC/FQ model, for temperatures of 10 90°C, finding quite good agreement with experiment. More recently, we have extended the method involving the quadratic electric field map for HOD/D20 [98] to HOD/H20 [52] and have calculated IR and unpolarized Raman line shapes. These line shapes, in comparison with experimental line shapes [12, 52], are shown in Fig. 7. Agreement between theory and experiment is excellent for both the IR and Raman. 

These authors also reported theoretical calculations of this frequency-dependent rotational relaxation. The theory of Auer et al. [98] using the quadratic electric field map, originally developed for HOD/D2O, was extended to the H0D/H20 system [52]. As before [38], the orientation TCF was calculated for those molecules within specified narrow-frequency windows (those selected in the experiment) at t = 0. TCFs for selected frequency windows, up to 500 fs, are shown in Fig. 8. One sees that in all cases there is a very rapid decay, in well under 50 fs, followed by a pronounced oscillation. The period of this oscillation appears to be between about 50 and 80 fs, which corresponds most likely to underdamped librational motion [154]. Indeed, the period is clearly longer on the blue side, consistent with the idea of a weaker H bond and hence weaker restraining potential. At 100 fs the values of the TCFs show the same trend as in experiment, although the theoretical TCF loses... 

The basic theory of Kohonen maps—and only this will be treated here—is mathematically simple. A typical Kohonen map consists of a rectangular (often quadratic) array of fields (squares, cells, nodes, neurons) with a typical size of 5 x 5 (25 fields) to 100 x 100 (10,000 fields). Each field k is characterized by a vector wk, containing the weights wki, wia, , with in being the number of variables of a multivariate data set X (Figure 3.18) the lengths of the weight vectors are, for instance,... 

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