Planar bifurcation

Fig. 11.2.4. Planar bifurcation of a saddle-node equilibrium state with 2 > 0.

Fig. 13.2.1. Planar bifurcation of a stable separatrix loop of a saddle with (Tq < 0.

The numbers on Fig. 7 refer to the number of unstable modes (corresponding to eigenvalues of the linearized form of (8) with positive real part) for shapes along segments of the families. Only the planar shape up the bifurcation point with the (lAe)>family and a portion of the (lAe)-family are stable to disturbances with the symmetry imposed by this sample size. 

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

The complexes formed by H5 bound to 01 were found to be slightly less stable than the new with H5 bound to 02, for both the guanidinium and the pyrolidinium complex. Both type of complexes feature bifurcated hydrogen bonds, with 02 bound to H5 and H6. The double binding is achieved with some cost to the linearity of the hydrogen bond. The most stable complexes have been found to be planar. 

Vol. 1480 F. Dumortier, R. Roussarie, J. Sotomayor, H. Zoladek, Bifurcations of Planar Vector Fields Nilpotent Singularities and Abelian Integrals. VIII, 226 pages. 1991. 

A. Andronov, E. Leontovich, I. Gordon and A. Mayer, Bifurcation Theory of Planar Dynamical Systems (Nauka, Moscow, 1967). 

In Chapter 3, two limit cycles played a prominent role. The first occurred in the planar system, representing oscillations in the simple food chain. Neither the stability nor the uniqueness of the limit cycle was established. Kuang [Kl] shows that the limit cycle is unique and asymptotically stable - for parameter values near where the rest point loses stability - by examining the Hopf bifurcation from the rest point. For other values of parameters, the uniqueness and stability questions remain open. 

In the same way as the a, -separation has been performed, one can proceed to a cr, rr-separation.52 This separation has been used to evaluate the aromaticity of organic molecules and clusters. An index of aromaticity was proposed using a scale based on the bifurcation analysis of the ELF constructed from the separated densities. In principle, the total ELF has no information about tt and cr bonds, it depends only on the total density. Hence, the ELF does not show clear differences between both kinds of bonds. However, the topological analysis over separated densities, ones formed by the rr-orbitals and the other ones formed by the cr-orbitals, yields the necessary information.52 Of course, this is possible only for the molecules which present the cr, tt symmetries, i.e. planar molecules. The bifurcation analysis of the news ELFW and ELFCT can be interpreted as a measure of the interaction among the different basins and chemically, as a measure of electron delocalization.45 In this way, the tt and a aromaticity for the set of planar molecules described in the Scheme 1 has been characterized.52... 

A large part of the computational work has been influenced by the introduction of curvilinear coordinates, designed to take advantage of the topography of potential surfaces. These coordinates allow for a smooth change from reactant to product conformations and in effect transform the rearrangement problem into the much simpler one of inelastic collisions. The various treatments have employed reaction-path (or natural collision) coordinates less restricted reaction coordinates atom-transfer coordinates, somewhat analogous to those used for electron-transfer and, for planar and spatial motion, bifurcation coordinates. 

To illustrate the general procedure let us consider collinear reactions of type A + BC - AB + C. This means that the three atoms move on a line, e.g. the x-axis, and furthermore that velocities are also along the same line. Indicating relative atomic distances by xAB, xBC and xCA, we introduce centre-of-mass coordinates, which for reactants are x = xBC and X, the distance from A to the centre of mass of BC. Similar coordinates could be defined for products. Because of restrictions in the type of motion it is not possible to simultaneously account for the B + CA rearrangement. To do this one must proceed to planar or spatial motion and, for example, introduce bifurcation coordinates. 

For planar motion the wavefunction in bifurcation coordinates was expanded as... 

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