A weak focus

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. 

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... 

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. 

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... 

A periodic orbit collapses into a weak focus (the length of the periodic orbit shrinks to zero while it approaches the bifurcation point). This condition coincides with the condition defining the boundary of an equilibrium state with a single pair of pure imaginary eigenvalues provided that the Lyapunov value Li e) < 0. 

C.5. 62. us give a general formula for the first Lyapunov value at a weak focus of the three-dimensional system... 

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