Rotation number

Figure 2. Schematic of typical data and consistent Poincare sections from the quasiperiodic regime of Rayleigh-B nard convection. The rotation number W (in arbitrary units) is plotted versus Rayleigh number R for two different values...

Because of the large number of rotational levels in the upper and lower states, the overlap between the exciting laser line and the dopp-ler broadened absorption profile may be nonzero simultaneously for several transitions (u", / ) (v, f) with different vibrational quantum numbers v and rotational numbers J. This means, in other words, that the energy conservation law allows several upper levels to be populated by absorption of laser photons from different lower levels. 

In the three-branch horseshoe, the periodic oibit 0 is hyperbolic with reflection and has a Maslov index equal to no = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index n = 2 [10]. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. 

There exists a quantity (the rotation number) which is independent of the particular stroboscopic section we take and characterizes the entire torus. For a radial parameterization of the invariant circle the rotation number is defined—and may be computed—as the limit... 

It is based on Denjoy s theorem, and ris the rotation number. This algorithm, implemented by Chan (1983) computes invariant circles with irrational rotation numbers. We may, of course, discretize and solve for the whole invariant surface and not just for a section of it. Instead of having to integrate the system equations, we will then be solving for a much larger number of unknowns resulting from the additional dimension we had suppressed in the shooting approach we used. 

The reactants were introduced simultaneously into milling vessels. After the vessels had been hermetically sealed, air was removed with a vacuum pump, and residual air was removed by several flushings with purified nitrogen milling was carried out in a nitrogen atmosphere. Milling was at a constant amplitude (4 mm) and frequency (1475 rpm, recorded by the rotation number of the electric motor). 

There are (2 / + 1 )mj levels for each value of j, so the Boltzmann distribution tells us that the total number of molecules with rotational number j is given by ... 

From these return times it is easy to obtain the intra-nephron rotation number (i.e., the rotation number associated with the two-mode behavior of the individual nephron)... 

Fig. 12.12 Internal rotation number as a function of the parameter a calculated from the single-nephron model. Inserts present phase projections for typical regimes. Note how the intra-nephron synchronization is maintained through a complete period-doubling cascade to chaos.

Let us consider the case of a = 30 corresponding to a weakly developed chaotic attractor in the individual nephron. The coupling strength y = 0.06 and the delay time T2 in the second nephron is considered as a parameter. Three different chaotic states can be identified in Fig. 12.16. For the asynchronous behavior both of the rotation numbers ns and n f differ from 1 and change continuously with T2. In the synchronization region, the rotation numbers are precisely equal to 1. Here, two cases can be distinguished. To the left, the rotation numbers ns and n/ are both equal to unity and both the slow and the fast oscillations are synchronized. To the right (T2 > 14.2 s), while the slow mode of the chaotic oscillations remain locked, the fast mode drifts randomly. In this case the synchronization condition is fulfilled only for one of oscillatory modes, and we speak of partial synchronization. A detailed analysis of the experimental data series reveals precisely the same phenomena [31]. 

Fig. 12.16 Full and partial synchronization of the fast and slow motions between two interacting nephrons (/ = 13.5 s, a = 30.0 and y = 0.06). Full synchronization is realized when both the fast nj and slow ns rotation numbers equal to 1. To the right in the figure there is an interval where only the slow modes are synchronized. The delay / in the loop of Henle for the second nephron is used as a parameter.

C2v Number of Coordinates Translation and Rotation Number of Vibrations... 

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

Here, as the basic frequency to describe resonant state of systems, we define the rotation number... 

Figure 1. Time series of local rotation numbers. K — 0.8, b — 0.002. 1 x 107 steps. Residence on rational numbers and intermittent transition among them are observed.

Figure 1 shows time series of local rotation numbers. The orbit remains for a while at a certain resonance, and it escapes to another resonance intermittently. 

The idea to investigate structures of resonances in frequency space was, to the authors knowledge, devised by Martens et al. [19] and then sophisticatedly implemented by Laskar and co-workers [13,20,21]. Instead of using FFT as their methods, we concentrate on a simple toy model and adapt rotation numbers as basic frequencies. By this way, we can easily compute basic frequency so that we can investigate global features of resonances and those dependence on parameters. 

To visualize the structures of resonances, we use the density plot in the frequency space to include information on time, as follows First, we compute the rotation numbers modulo 1 over a finite time interval from trajectories to... 

With the representations of these resonance layers in mind, we clarify quantitative differences between isolated resonances and overlapped resonances, by examining residence time distributions p(f) at each resonance layer. Since there are fluctuations in local rotation numbers due to the finite time average, we set some threshold W for each resonance condition, so that we compute the... 

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