Center manifold points

Transport. We need now to construct the NHIM, its stable/unstable manifolds, and the center manifold. Let P be the main relative equilibrium point. The first task is to find the short periodic orbits lying above P in energy. These p.o. are unstable. We did so by exploring phase space at energies 4, 10, and 14 cm above E (1 atomic unit = 2.194746 x 10 cm ). It is not possible to go much higher in E, since the center manifold disappears shortly above E + 14cm , because of the structure of the potential energy surface. 

Moreover, under the assumption of pure point critical spectrum, a center manifold description of the associated meandering, drifting, and resonance phenomena has been achieved see [22], [73], and the survey [21]. We give a brief exposition in section 3.2.1. Specific calculations, however, require disproportionate computational effort to account for the two-dimensional time dependent problem in large (theoretically unbounded) domains. 

Prom [72] we recall the existence of a center manifold Ai associated to a rigidly rotating reference spiral, alias a relative eqnilibrium SE 2)u see also (3.1)-(3.7) above. In fact, the center manifold M. accounts for all solutions which remain in a neighborhood of the relative equilibrium SE 2)ut, for all positive and negative times. In particular, all bifurcations due to point spectrum on the imaginary axis of the linearization L from... 

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS. ... 

Application of the center manifold theorem [1] to system (1) leads to a statement of very great generality. With its aid it is possible to characterize the topology of the ensemble of all solutions of system (1). This is done in a phase space ft of n+1 dimensions consisting of the union a U z. The key to the understanding of choking is in the identification of the singular points of system (1) whose coordinates in phase space ft are solutions o, z of the simultaneous equations... 

Center Manifolds. The Center Manifold Theorem (see Carr (1981)) states that all branches of stationary and periodic states in a neighborhood of a bifurcation point are embedded in a sub-manifold of the extended phase space X M that is invariant with respect to the flow generated by the ODE (2.1). All trajectories starting on this so-called center manifold remain on it for all times. All trajectories starting from outside of it exponentially converge towards the center manifold. Specifically, static bifurcations are embedded in a two dimensional center manifold, whereas center manifolds for Hopf bifurcations are three dimensional. Figures 2.1 and 2.2 summarize the geometric properties of the flows inside a center manifold in the case of saddle-node and Hopf bifurcations, respectively. 

Normal Forms. In order to model essential nonlinear properties of a given flow nearby a critical point one can focus on a center manifold that locally contains all critical points, steady states, and periodic orbits. In the case of static bifurcations it turns out that a one dimensional parameter dependent ODE suffices to describe the dynamics inside the center manifold. For... 

Transversely to the center manifold, another invariant manifold passes through the point 0(0,0). It is called strongly stable and, as usual, we denote it by Its equation is given by a = y), where (y) vanishes at... 

In a neighborhood of the point O there exists a -smooth change of variables which straightens both the invariant foliation and the center manifold so that the system in the new variables assumes the following standard form... 

As shown in Chap. 5, the critical fixed point 0(0,0) lies in an invariant C" -smooth center manifold defined by the equation y = (x), where vanishes at the origin along with its first derivative. Moreover, the following reduction theorem holds ... 

Thus, the stability of the fixed point of the original map (10.1.1) is equivalent to the stability of the fixed point of the map (10.1.6) in the center manifold, which we state formally as follows... 

If all Lyapunov values vanish and the map is analytic, then the center manifold is analytic too and it consists of fixed points (Fig. 10.2.8). Observe that if the map has the form... 

From (10.2.5)-(10.2.7) we can conclude that the analytic curve y = (t> x) consists entirely of fixed points and, therefore, is an invariant (center) manifold. 

If all Lyapunov values are equal to zero and the system is analytic, then the center manifold is also analytic, and all points on it, except O, are periodic of period two. This means that for the system of differential equations there exists a non-orientable center manifold which is a Mobius band with the cycle L as its median and which is filled in by the periodic orbits of periods close to the double period of L (see Fig. 10.3.2). 

If Lk < 0, then for the original multi-dimensional map (10.4.1), the fixed point is also a stable focus. Moreover, its leading manifold coincides with the center manifold. This means that all positive semi-trajectories, excluding those in the non-leading manifold tend to O along spirals which are... 

The remarkable feature of this bifurcation in the case of periodic orbits of autonomous systems of differential equations is that the center manifold of the periodic orbit L corresponding to the fixed point O of the Poincare map is a Mobius band. The orbit itself is the mean line of the Mobius band, and consequently a new orbit that bifurcates from L must wind twice around L as shown in Fig. 11.4.5. It is quite clear that the period of the new orbit is nearly the double period of L. Consequently, this bifurcation is called... 

The above theorem is related to the map on the center manifold. Reconstructing the behavior of trajectories of the original map (11.6.2) is relatively simple. Here, if L < 0, then the fixed point is stable when /i < 0. When /i > 0 it becomes a saddle-focus with an m-dimensional stable manifold (defined by T = 0) and with a two-dimensional unstable manifold which consists of a part of the plane y = 0 bounded by the stable invariant curve C,... 

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

It follows from the linear stability analysis (see Sec. C.2) that the (a,6)-parameter plane has a codimension-two point corresponding to an equilibrium state with characteristic exponents (0, ztia ). Therefore, the dimension of the center manifold in such a case must be equal to 3 at least. For the complete accoimt on this bifurcation the reader is referred to [51, 64]. Below, we will give its brief outline. 

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. 

From a geometric point of view, the autonomous fixed point is the organizing center for the hierarchy of invariant manifolds. From a technical point of view, it is also the expansion center around which all Taylor series expansions are carried out. If the TS trajectory is to take over the role of the fixed point, this observation suggests that it be used as a time-dependent coordinate origin. We therefore introduce the relative coordinates... 

We have outlined how the conceptual tools provided by geometric TST can be generalized to deterministically or stochastically driven systems. The center-piece of the construction is the TS trajectory, which plays the role of the saddle point in the autonomous setting. It carries invariant manifolds and a TST dividing surface, which thus become time-dependent themselves. Nevertheless, their functions remain the same as in autonomous TST there is a TST dividing surface that is locally free of recrossings and thus satisfies the fundamental requirement of TST. In addition, invariant manifolds separate reactive from nonreactive trajectories, and their knowledge enables one to predict the fate of a trajectory a priori. 

The Molecular Surface (MS) first introduced by Richards (19) was chosen as the 3D space where the MLP will be calculated. MS specifically refers to a molecular envelope accessible by a solvent molecule. Unlike the solvent accessible surface (20), which is defined by the center of a spherical probe as it is rolled over a molecule, the MS (19), or Connolly surface (21) is traced by the inwardfacing surface of the spherical probe (Fig. 2). The MS consists of three types of faces, namely contact, saddle, and concave reentrant, where the spherical probe touches molecule atoms at one, two, or three points, simultaneously. Calculation of molecular properties on the MS and integration of a function over the MS require a numerical representation of the MS as a manifold S(Mk, nk, dsk), where Mk, nk, dsk are, respectively, the coordinates, the normal vector, and the area of a small element of the MS. Among the published computational methods for a triangulated MS (22,23), the method proposed by Connolly (21,24) was used because it provides a numerical presentation of the MS as a collection of dot coordinates and outward normal vectors. In order to build the 3D-logP descriptor independent from the calculation parameters of the MS, the precision of the MS area computation was first estimated as a function of the point density and the probe radius parameters. When varying... 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

The coordinates QJ generate an F-dimensional vector space (or manifold) M-the configuration space of the nuclei. The positions of the nuclei in the configuration space M are given by a single point, the system point Q = QJ. Analogously, let the positions of the electrons in the center-of-mass system be given by the coordinates q = qu, u — 1,..., 3n. 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

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