Structurally stable saddle

Stable adsorption complexes are characterized by local minima on the potential energy hypersurface. The reaction pathway between two stable minima is determined by computation of a transition state structure, a saddle point on the potential energy hypersurface, characterized by a single imaginary vibrational mode. The Cartesian displacements of atoms that participate in this vibration characterize movements of these atoms along the reaction coordinate between sorption complexes. 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

In contrast to the predictions of INDO calculations, however, ab initio calculations232 (3-21 G basis set) on CH2=CH2-, CH CHF-, CH2=CF2 and CF2=CF2- have predicted that planar structures are not stable minima for these radical anions. The planar structures relax to the anti structures upon minimization, and the stabilization due to non-planar distortions, arising from a mixing of the 7r - and the high lying cr -orbitals, increases upon fiuorination. For these four molecules the 90° twisted structures are saddle points, and the CH2CF2 radical anion was predicted to adopt the chair structure. 

Hyperbolic fixed points also illustrate the important general notion of structural stability. A phase portrait is structurally stable if its topology cannot be changed by an arbitrarily small perturbation to the vector field. For instance, the phase portrait of a saddle point is structurally stable, but that of a center is not an arbitrarily small amount of damping converts the center to a spiral. 

Notice that for a 0, the phase portrait has a different topological character the saddles are no longer connected by a trajectory. The point of this exercise is that the phase portrait in (a) is not structurally stable, since its topology can be changed by an arbitrarily small perturbation a. ... 

If eigenvalues are real and different, then a given dynamical system is locally (in the vicinity of a stationary point) equivalent to a certain structurally stable gradient system (this is an unstable node when 2, > 0, X2 > 0 a saddle when A, > 0, X2 < 0 or Xt < 0, X2 > 0 a stable node when Xx < 0, X2 > 0). In the remaining cases a dynamical system is not locally equivalent to a gradient system. 

The saddle node catastrophe and the Hopf bifurcation may be shown to be structurally stable. Certain additional conditions (see Sections 5.5.2.2, 5.5.2.3) are imposed on the transcritical bifurcation and the pitchfork bifurcation. The system is structurally stable under perturbations not disturbing these additional conditions on the other hand, when arbitrary... 

Fig. 1. Phase plane representations of structurally stable dynamics found in chemical and biological systems, (a) Single stable steady state, which is approached in oscillatory fashion, (b) Limit < cle attractor to which all trajectories tend in the limit r oo. (c) Two stable steady states, whidi have different basins of attraction. The initial conditions determine which steady state is reached. There is also a saddle point.

This result shows that an individual trajectory cannot give an adequate image of chaotic oscillations. Looking ahead we note that all imclosed Poisson-stable trajectories in structurally stable systems are, in fact, unstable, or more precisely, of the saddle type. 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

Fig 7.5.2. A structurally stable heteroclinic connection between two saddles in... 

Recall that a fixed point 0 x = xq) is called structurally stable if none of its characteristic multipliers, i.e. the roots of the characteristic equation (7.5.2), lies on the unit circle, A topological type (m,p) is assigned to it, where m is the number of roots inside the unit circle and p is that outside of the unit circle. If m = n (m = 0), the fixed point is stable (completely unstable). The fixed point is of saddle type when m 0,n. The set of all points whose trajectories converge to xq when iterated positively (negatively) is called the stable (unstable) manifold of the fixed point and denoted by Wq Wq). In the case where m = n, the attraction basin of O is Wq. If the fixed point is a saddle, the manifolds Wq and Wq are C -smooth embeddings of and MP in respectively. 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

Note that in many special cases attention is restricted to the study of the smaller spaces of systems, e.g. systems with some specified symmetries, Hamiltonian systems, etc. In view of that, the notion of structural stability in, say, Hamiltonian systems with one-degree-of-freedom becomes completely meaningful. So, for example, equilibrium states such as centers and saddles of such systems, become structurally stable. Moreover, if there are no heteroclinic cycles containing different saddles, we can naturally distinguish such systems as rough in the set of all systems of the given class. 

Note that only cases (1) and (2) correspond to structurally stable systems the other cases are non-rough. In essence, a bifurcation of a homoclinic-8 with a negative saddle value is an internal bifurcation in the Morse-Smale class. 

Here there is one sign change in the first column, i.e. H( ) has one root in the right open half-plane. Let us count the number of purely imaginary roots 2(p — 1 — Z) = 2(2 — 1 — 1) = 0. Thus, the equilibrium state O is structurally stable, and its topological type is saddle (3,1). ... 

We have q = — 7a /36 < 0 at i2 = 0. This means that the point at the origin cannot have a pair of purely imaginary eigenvalues. Thus, it is always structurally stable when (a, b) 0. In accordance to the above classification table, its topological type is a saddle with a two-dimensional stable manifold, and a one-dimensional unstable manifold. 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

A theoretical study at a HF/3-21G level of stationary structures in view of modeling the kinetic and thermodynamic controls by solvent effects was carried out by Andres and coworkers [294], The reaction mechanism for the addition of azide anion to methyl 2,3-dideaoxy-2,3-epimino-oeL-eiythrofuranoside, methyl 2,3-anhydro-a-L-ciythrofuranoside and methyl 2,3-anhydro-P-L-eiythrofuranoside were investigated. The reaction mechanism presents alternative pathways (with two saddle points of index 1) which act in a kinetically competitive way. The results indicate that the inclusion of solvent effects changes the order of stability of products and saddle points. From the structural point of view, the solvent affects the energy of the saddles but not their geometric parameters. Other stationary points geometries are also stable. 

The question of methanol protonation was revisited by Shah et al. (237, 238), who used first-principles calculations to study the adsorption of methanol in chabazite and sodalite. The computational demands of this technique are such that only the most symmetrical zeolite lattices are accessible at present, but this limitation is sure to change in the future. Pseudopotentials were used to model the core electrons, verified by reproduction of the lattice parameter of a-quartz and the gas-phase geometry of methanol. In chabazite, methanol was found to be adsorbed in the 8-ring channel of the structure. The optimized structure corresponds to the ion-paired complex, previously designated as a saddle point on the basis of cluster calculations. No stable minimum was found corresponding to the neutral complex. Shah et al. (237) concluded that any barrier to protonation is more than compensated for by the electrostatic potential within the 8-ring. 

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