Fast-drift plane

Let us now set for a moment R (p,q) = 0. Then, according to the general theory discussed in Section 2.5, the Hamiltonian in normal form possesses n—dim M independent first integrals of the form geometrical considerations we conclude that any orbit with initial point po G V lies on a plane n wj(p0) through Po and parallel to M we shall call this plane the plane of fast drift. This is true in the coordinates of the normal form. If we look at the original coordinates then we must take into account the deformation due to the canonical transformations —as we already remarked while discussing the case of an elliptic equilibrium. Moreover, we must consider also the noise due to the remainder, but in this case too we have = 0(er), so that the noise causes only a slow drift that becomes comparable with the deformation only after a time T(e) l/er. 

The conclusion is illustrated in Figure 4 until the orbit remains in the non-resonance domain V it lies in a small neighborhood of the plane of fast drift nx(po)- However, this is not enough to assure the long-time preservation of the actions, because the analytic theory does not assure that the orbit will be confined for a long time in V. The question is what happens if the orbit leaves the non-resonance domain ... 

Figure 4 The local dynamics in a non-resonance domain. The orbit lies in a neighborhood of the plane of fast drift.

If tlie recovery is very fast, then part of it may be masked by the acquisition parameters. To correct for this, try using a scope with two separate lasers - one for imaging and one for bleaching. If a separate laser is not available, then try increasing the speed of data acquisition by only collecting data from the ROI versus the entire cell. However, it is important to note that this method sacrifices information about overall photobleaching and/or focal plane drifts which may impact your data. Alternatively, the size of the ROI can be increased. 

The plane front is the most favourable type for fast chemical reactions with low MW compounds. This mode is intermediate between the torch and drift, due to the strongly developed turbulence in reactant mixing zone 1 (coaxial supply) and zone 2 (radial supply). The plane front is characterised by the homogeneity of the reaction flow composition along the reactor radius, which is typical of the quasi-plug flow mode in turbulent flows. 

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