Weak focus

Unfortunately this condition is not sufficient, and it is shown that a similar condition exists also in the case of the weak focus to distinguish between the two requires the introduction of approximations of high order. 

Figure 1. Schematic illustration of the laser-vaporization supersonic cluster source. Just before the peak of an intense He pulse from the nozzle (at left), a weakly focused laser pulse strikes from the rotating metal rod. The hot metal vapor sputtered from the surface is swept down the condensation channel in dense He, where cluster formation occurs through nucleation. The gas pulse expands into vacuum, with a skinned portion to serve as a collimated cluster bean. The deflection magnet is used to measure magnetic properties, while the final chaiber at right is for measurement of the cluster distribution by laser photoionization time-of-flight mass spectroscopy.

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

In most of the study lakes organic content is poorly correlated with lake depth, a feature that may be partially attributed to the fact that fine-grained sediments are only weakly focused in small shallow lakes (6, 18). Indeed, sediments do become somewhat more organic with depth in the two deepest lakes, Thrush and Kjostad. The perception of uniform sediment composition also results from sampling bias, because few littoral cores were actually retained for analysis. Visibly inorganic sediments were assumed to represent nondepositional sites for purposes of calculating a Hg flux and were simply mapped and discarded. 

Fig. 16.2. An example of an all-reflective, non-common-path geometry. Light is delivered by a weakly focused beam, and large-angle scattering is gathered and collimated by the paraboloidal reflector see [2] for an application and further references...

The muon g — 2 value has been determined in a series of experiments at CERN [45,46]. The primary purpose of the new muon g — 2 experiment at Brookhaven National Laboratory is to improve the precision of the experiment by about a factor 20 and verify the presence of the electroweak effect which has been evaluated to two loop orders in the Standard Model. In this experiment, polarized muons from pion decays are captured in a storage ring with a uniform magnetic field and a weak-focusing electric quadrupole field. For a muon momentum of 3.09 GeV/c and 7 = 29.3 the muon spin motion is unaffected by the electric quadrupole field and the difference frequency uia is given by... 

Falconer, B.T. and Murphy, P. (2005), The Can-Do Attitude Strength or Weakness , Focus - Special Human Factors Edition, Canberra, Australia Department of Defence. 

The laser beams are weakly focused on the molecular beam by a 400 mm quartz lens, and each pulse reaches (in the femtosecond experiments) a peak power of about 0.5GWcm. Photoionized molecules and clusters are guided by electrical lenses into the axis of a quadrupole mass spectrometer (Balzers QMG 420). The ions are mass-selected with a resolution mfAm > 200, which is sufficient to distinguish between, for example, isotopic species and... 

Fia 13.11. Longitudinal and transverse forces exerted on a neutral atom in a weakly focused Gaussian laser beam [13.16]... 

Such equilibrium state is called a weak focus. It is stable if Li < 0 and unstable if L > 0. 

If Z/fc > 0, the origin is an unstable equilibrium state because trajectories starting close to it spiral away as time increases. For the two-dimensional system (9.3.1) the point O is called an unstable complex weak) focus. 

Fig. 9.3.1. A stable (Lk < 0) weak focus M. When L. > 0, a trajectory leaves a neighborhood of the origin along a counter clock-wise spiral.

We have seen in the previous sections that the qualitative behavior of a strongly resonant critical fixed point differs essentially from that of a non-resonant or a weakly resonant one. It is therefore natural to ask the question what happens at a strongly resonant point as the frequency varies In particular, in the case of the resonance a = 27t/3 the fixed point is a saddle with six separatrices in general, but when an arbitrarily small detuning is introduced the point becomes a weak focus (stable or unstable, depending on the sign of the first Lyapunov value). The question we seek to answer is how does the dynamics evolve before and after the critical moment ... 

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

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