Stable invariant manifold

Up to moderately high energy ( 179%) of the activation barrier for reactant product in the Are isomerization reaction, the fates of most trajectories can be predicted more accurately by Eq. (11) as the order of perturbation calculation increases, except just in the vicinity of the (approximate) stable invariant manifolds (e.g., see Eig. 5), and that the transmission coefficient K observed in the configurational space can also be reproduced by the dynamical propensity rule without any elaborate trajectory calculation (see Eig. 6). Our findings indicate that almost all observed deviations from unity of the conventional transmission coefficient k may be due to the choice of the reaction coordinate whenever the k arises from the recrossings, and most transitions in chemical... 

The stable invariant manifolds of equilibrimn states and periodic orbits may have common points with the unstable manifolds. Clearly, if a point xq is such a common point of two invariant manifolds, then the trajectory x = xq) belongs to both manifolds entirely. In the simplest case, O = Wq n W, i.e. the stable and imstable manifolds of an equilibrium state intersect at only trajectory, which is the equilibrium state itself. 

In the new coordinates, the strong-stable invariant manifold Wff is the surface x = 0 the node region C/ now corresponds to small negative x and the saddle region t/ corresponds to small positive x. 

Two alternatives present themselves in formulating algorithms for the tracking of segments of stable and unstable manifolds. The first involves observing the initial value problem for an appropriately chosen familv of initial conditions, henceforth referred to as simulation of invariant manifolds. A second generation of algorithms for the computation of invariant manifolds involves numerical fixed point techniques. 

A typical trajectory has nonzero values of both P and Q. It is part of neither the NHIM itself nor the NHIM s stable or unstable manifolds. As illustrated in Fig. la, these typical trajectories fall into four distinct classes. Some trajectories cross the barrier from the reactant side q < 0 to the product side q > 0 (reactive) or from the product side to the reactant side (backward reactive). Other trajectories approach the barrier from either the reactant or the product side but do not cross it. They return on the side from which they approached (nonreactive trajectories). The boundaries or separatrices between regions of reactive and nonreactive trajectories in phase space are formed by the stable and unstable manifolds of the NHIM. Thus once these manifolds are known, one can predict the fate of a trajectory that approaches the barrier with certainty, without having to follow the trajectory until it leaves the barrier region again. This predictive value of the invariant manifolds constitutes the power of the geometric approach to TST, and when we are discussing driven systems, we mainly strive to construct time-dependent analogues of these manifolds. 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

In the absence of damping (and in units where ( b = 1), the invariant manifolds bisect the angles between the coordinate axes. The presence of damping destroys this symmetry. As the damping constant increases, the unstable manifold rotates toward the Agu-axis, the stable manifold toward the A<7u-axis. In the limit of infinite damping the invariant manifolds coincide with... 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

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. 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

Wiggins et al. [22] pointed out that one can always locally transform a Hamiltonian to the form of Eq. (1.38) if there exists a certain type of saddle point. Examination of the associated Hamilton s equations of motion shows that q = Pn = 0 is a fixed point that defines an invariant manifold of dimension 2n — 2. This manifold intersects with the energy surface, creating a (2m — 3)-dimensional invariant manifold. The latter invariant manifold of dimension 2m — 3 is an excellent example of an NHIM. More interesting, in this case the stable and unstable manifolds of the NHIM, denoted by W and W ,... 

Figure 4. A schematic portrait of the stable and unstable invariant manifolds and the phase-space flows on (4i(p,q),Pi(p,q)).

C. Normally Hyperbolic Invariant Manifolds (NHIMs) and Their Stable and Unstable Manifolds... 

This is also the Hamiltonian of the activated complex. We will encounter it in Eq. (23) with the customary symbol H. Regardless of its stability properties or the size of the nonlinearity, Eq. (12) is always an invariant manifold. However, we are interested in the case when it is of the saddle type with stable and unstable manifolds. If the physical Hamiltonian is of the form of Eq. (1), then a preliminary, local transformation is not required. The manifold (12) is invariant regardless of the size of the nonlinearity. Moreover, it is also of saddle type with respect to stability in the transverse directions. This can be seen by examining Eq. (1). On qn = Pn = 0 the transverse directions, (i.e., q and p ), are still of saddle type (more precisely, they grow and decay exponentially). 

Normally, hyperbolic invariant manifolds persist under perturbation [22]. If we are in the setting where the form of Eq. (1) must hrst be obtained by applying Normal Form theory, then we are restricted to a sufficiently small neighborhood of the equilibrium point. In this case the nonlinear terms are much smaller than the linear terms. Therefore, the sphere present in the linear problem becomes a deformed sphere for the nonlinear problem and still has (2n — 2)-dimensional stable and unstable manifolds in the (2n — l)-dimensional energy surface since normal hyperbolicity is preserved under perturbations. 

In Fig. 6, the dynamics on can be decomposed into the movement along the normal directions and the flow on the manifold Mg. Note that the time development of b does not affect the movement of the base points. Suppose two points PI and PI on with the same base point yl. Then, the orbit from PI through P2 reaching P3 and the one from PI through P2 reaching P3 are projected to the same movement of the base points on from yl through y2 reaching y3. In other words, the three-dimensional invariant manifold Wl consists of two-dimensional invariant manifolds that correspond to the movements of the base points. Then, in mathematics, we say that the three-dimensional stable manifold is foliated by two-dimensional leaves. This structure is called foliation [28,30]. 

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