Chaos frequency parameters

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

Figure 1. Comparison at identical parameter values of experimental and quantum-mechanical values for the microwave field strength for 10% ionization probability as a function of microwave frequency. The field and frequency are classically scaled, u>o = and = q6, where no is the initially excited state. Ionization includes excitation to states with n above nc. The theoretical points are shown as solid triangles. The dashed curve is drawn through the entire experimental data set. Values of no, nc are 64, 114 (filled circles) 68, 114 (crosses) 76, 114 (filled squares) 80, 120 (open squares) 86, 130 (triangles) 94, 130 (pluses) and 98, 130 (diamonds). Multiple theoretical values at the same uq are for different compensating experimental choices of no and a. The dotted curve is the classical chaos border. The solid line is the quantum 10% threshold according to localization theory for the present experimental conditions.

Two-frequency irradiation of a two-level atom was proposed by Pomeau et al. (1986). As compared with one-frequency irradiation, a rapid decay of correlations, indicative of true quantum chaos, was observed. However, for the particular choice of parameters in the paper by Pomeau et al. (1986), Badii and Meier (1987) were able to demonstrate that the response is not chaotic, but quasi-periodic, albeit on a very long time... 

Chaos does not occur as long as the torus attraaor is stable. As a parameter of the system is varied, however, this attractor may go through a sequence of transformations that eventually render it unstable and lead to the possibility of chaotic behavior. An early suggestion for how this happens arose in the context of turbulent fluid flow and involved a cascade of Hopf bifurcations, each of which generate additional independent frequencies. Each additional frequency corresponds to an additional dimension in phase space the associated attractors are correspondingly higher dimensional tori so that, for example, two independent frequencies correspond to a two-dimensional torus (7 ), whereas three independent frequencies would correspond to a three-dimensional torus (T ). The Landau theory suggested that a cascade of Hopf bifurcations eventually accumulates at a particular value of the bifurcation parameter, at which point an infinity of modes becomes available to the system this would then correspond to chaos (i.e., turbulence). 

J. Chao, R. C. Wilhoit and B. J. Zwolinski, Ideal gas thermodynamic properties of ethane and propane , J. Phys. Chem. Ref. Data, 2, 427 (1973). Review and evaluation of structural parameters (including vibrational frequencies and internal rotation properties) tabulation of thermodynamic properties [C°, S°, H° — H°), (H° — H )/T, - G°-Hl)/T, AfG°,AfH°, logK ] for 0< T (K)< l500 calculated by statistical thermodynamic methods [rigid-rotor harmonic oscillator (RRHO) approximation]. 

If this expected photoemission really takes place, the resultant spectra should reflect the nonhnear dynamics of nonadiabatic vibrational motion under an external field, which is similar to classical driven oscillators such as a forced Duffing oscillator [156, 239]. Therefore various nonlinear phenomena such as limit cycle, frequency locking, and chaos (1.5-dimensional chaos) [156, 239] can be expected, which would be intrinsically originated from the quantmn dynamics. Furthermore, one may be able to control the frequency and amplitude of the photoemission by varying the laser parameters applied. It may be possible to utilize the photoemission as a new optical somce and also as finger-print signals to identify molecular species and/or molecular states. In this section we illustrate the appearance of such... 

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