Structural change bifurcation mechanism

The examples previously discussed with reference to the structure diagram demonstrated the existence of two kinds of catastrophe points, called bifurcation and conflict points. Both types of instabilities were illustrated in terms of the behaviour observed for molecular charge distributions. What we now show is that the existence of these two kinds of catastrophes and just these two, is a direct consequence of a theorem of structural stability stated by Palis and Smale in 1970. This theorem predicts what are the two basic mechanisms for structural change in a chemical system. 

One finds that the structure diagrams obtained for F(r X) and p(r X) for the HjO system are homeomorphic in the sense that both exhibit an identical partitioning of the control space yielding the same sets of structures, both stable and, as illustrated in Fig. 3.13, unstable. In addition to finding the same sets of structures for both fields, it has been found that their mechanisms of structural change are also the same. Thus, the bifurcation and conflict... 

Figure 9 Maps of the gradient vector field of the electron density along the symmetrical dissociation path for water illustrating the bifurcation mechanism of structural change. Also shown for structures a. b. and c are profiles of the electron density along the C2 symmetry axis. In a there are no cps along this axis in b there is a single degenerate cp where both first and second derivatives of p(r ) varysh in c there are two stable cps, the maximum associated with the H-H bond cp, the minimum with the ring cp...

These examples have identified two types of catastrophe points, a distinction that arises as a corollary of a theorem on structural stability. This theorem, when used to describe structural changes in a molecular system, states that the structure associated with a particular geometry X in nuclear configuration space is structurally stable if p r X) has a finite number of cps such that (i) each cp is nondegenerate (ii) the stable and unstable manifolds of any pair of cps intersect transver-sally. The immediate consequence of this theorem is that a structural instability can be established solely through either of two mechanisms in the bifurcation mechanism the charge distribution exhibits a degenerate cp, while the conflict mechanism is characterized by the nontransversal intersection of the stable and unstable manifolds of cps in p(r X). 

In down-wash configuration, the flame is established in the wake of the burner tube. A recirculation vortex in the wake of a burner tube appears as a flame sheet. When R is further reduced, the flame tip is severely deflected by the crossflow. A small recirculation bubble was observed by Huang and Chang [16] atR = 0.04. For a value of R between 0.1 and 1, the impact of a cross-flow stream is dominant. An axisymmetric tail flame forms downstream of the recirculation vortex and the flame widens. This structure is characterized by several features such as flickering and bifurcation. In jet-dominated mode, the recirculation vortex disappears and only the tail part remains attached to the burner. The transition from crossflow-dominated to jet-dominated conditions occurs from 1 = 1 to 3. For R>3, the effect of crossflow becomes negligible the jet fluid mechanics dictate the flame characteristics. For R > 10, the flame detaches from the burner and stabilizes above the exit plane of the burner tip. Depending upon the jet exit velocity and burner diameter, the flame is either attached to the burner tip or stabilizes as a lifted flame until it blows out. 

The immediate consequence of the theorem is that a structural instability can be established through only one of two possible mechanisms which correspond to the bifurcation and conflict catastrophes. A change in molecular structure—the making and breaking of chemical bonds—can only be caused by the formation of a degenerate critical point in the electronic charge distribution or by the attainment of an unstable intersection of the submanifolds of bond and ring critical points. 

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