Centre-manifold approach

The effect of additive as well as multiplicative noise by the centre-manifold approach was studied by Knobloch Wiesenfeld (1983). The main point of their procedure is that in the vicinity of bifurcation points a reduced Fokker-Planck equation is sufficient to describe the dynamics (after a short relaxation time). 

The centre-manifold approach is a powerful technique for reducing dimensions at a bifurcation point (Kelley, 1967). 

Adding a term due to Gaussian white noise to the second equation of (5.176) and adopting the centre-manifold approach it can be calculated that the stationary solution of the reduced Fokker-Planck equation is... 

As described earlier, the lifetimes and quantum yields of emissive Ln complexes vary dramatically due to the extremely sensitive nature of the 4/-centred excited states to 0-H, N-H and C-H vibrational manifolds, which can provide efficient, non-radiative deactivation pathways the efficiency of energy transfer between the antenna and lanthanide ion also determines overall quantum yields. A classical approach to maximising the emissivity of Ln complexes is to therefore inhibit the approach of water solvent to the inner coordination sphere (and where q denotes the number of coordinated solvent molecules) high denticity, metal ion encapsulating ligands with hydrophobic peripheries can achieve this very effectively, reducing q to zero [4]. 

---