Normal hyperbolicity

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer, New York, 1994. 

A parameter used to assess the degree of cooperativity exhibited by an enzyme ". Rs equals the ratio of [S]o.9/ [S]o.o9 that is, the ratio of the substrate concentration needed for 90% saturation divided by the substrate concentration needed for 10% saturation. For a normal, hyperbolic, noncooperative curve, Rs equals 81. Thus, positively cooperative systems will have an Rs ratio less than 81, whereas negatively cooperative systems will have values larger than 81. The ratio is insensitive to the shape of the curve and does not address any questions concerning the substrate concentration range between the 10% and 90% points. 

In the absence of activators AMP aminohydrolase from brain (149), erythrocytes (143, 150), muscle (145), and liver (128) gave sigmoid curves for velocity vs. AMP concentration which were hyperbolic after the addition of monovalent cations, adenine nucleotides, or a combination of monovalent cations and adenine nucleotides. For the rabbit muscle enzyme (145), addition of K+, ADP, or ATP produced normal hyperbolic saturation curves for AMP as represented by a change in the Hill slope nH from 2.2 to 1.1 Fmax remained the same. The soluble erythrocyte enzyme and the calf brain enzyme required the presence of both monovalent cations and ATP before saturation curves became hyperbolic. In contrast, the bound human erythrocyte membrane enzyme did not exhibit sigmoid saturation curves and K+ activation was not affected by ATP (142). 

An unstable periodic orbit is one-dimensional, being of dimension two less than the energy surface in systems with two DOFs. In an n-DOF system the energy surface is of dimension 2n — 1. In such systems, Wiggins showed that the analog of unstable periodic orbits is the so-called normally hyperbolic invariant manifold (NHIM) of dimension 2n — 3 [20,21]. Trajectories slightly displaced from an NHIM can be analyzed using a many-dimensional stability analysis. The... 

Recently, Wiggins et al. [15] provided a firm mathematical foundation of the robust persistence of the invariant of motion associated with the phase-space reaction coordinate in a sea of chaos. The central component in RIT that is, unstable periodic orbits, are naturally generalized in many DOFs systems in terms of so-called normally hyperbolic invariant manifold (NHIM). The fundamental theorem on NHIMs, denoted here by M, ensures [21,53] that NHIMs, if they exist, survive under arbitrary perturbation with the property that the stretching and contraction rates under the linearized dynamics transverse to jM dominate those tangent to M. Note that NHIM only requires that instability in either a forward or backward direction in time transverse to M is much stronger than those tangential directions of M, and hence the concept of NHIM can be applied to any class of continuous dynamical systems. In the case of the vicinity of saddles for Hamiltonian problems with many DOFs, the NHIM is expressed by a set of all (p, q) satisfying both q = p = Q and o(Jb) + En=i (Jb, b) = E, that is. 

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

Moreover, the intersection of the center manifold with an energy shell yields an NHIM. The NHIM, which is a (2n — 3)-dimensional hypersphere, is the higher-dimensional analog of Pechukas PODS. Because this manifold is normally hyperbolic, it will possess stable and unstable manifolds. These manifolds are the 2n — 2)-dimensional analogs of the separatrices. The NHIM is the edge of the TS, which is a (2n — 2)-dimensional hemisphere. 

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. 

This very simple Hamiltonian is at the basis of the whole TS approach. It generalizes easily into many dimension (Section IV), is a good basis for perturbation theory [4], and is also the basis for numerical schemes, classical and semiclassical. The inclusion of angular momentum implies that some ingredients must be added (see Section V). Let us thus describe how this very simple, linear Hamiltonian supports normally hyperbolic invariant manifolds (NHIMs see Section IV for a proper discussion) separatrices and a transition state. 

We shall make more use of the notion of normally hyperbolic invariant manifold (NHIM). This invariant surface is the n-DOF generalization of the periodic orbit dividing surface, even if originally defined in a much more general framework (a bibliography may be found in Ref. 24). Its correct definition is put forward in Section IV.A and is used in all examples coming thereafter. 

In the liner approximation, we see thus that the NHIM is made of periodic/ quasi-periodic orbits, organized in the usual tori characteristic of the integrable systems. Because the NHIM is normally hyperbolic, each point of the sphere has stable/unstable manifolds attached to it. This situation is exactly parallel to the one described earlier for PODS. The equation for it is... 

However, there is another, more specific, yet more interesting, way to portray the sphere. Recalling that not only energy but also angular momentum A is conserved, the sphere S3 is also parameterized with the three quantities y(4, A, 4>), with 4> the angle conjugated to A. This view is particularly useful for the construction of the normally hyperbolic manifolds. 

First, in order to simplify the description of the dynamics we separate the whole system, locally in the phase space, into two parts based on a gap in characteristic time scales. This is done using the concept of normally hyperbolic invariant manifolds (NHIMs) [4-8]. Here, the characteristic time scales are estimated as the inverses of the absolute values of the local Lyapunov exponents [5,6]. Then, the Fenichel normal form offers a simplified description of the local dynamics near a NHIM [7]. 

Here, we mention only two possibilities, though we could have other cases. The hrst is that the condition of normal hyperbolicity breaks down for some NHIMs. Then, what happens to those NHIMs Do they bifurcate into other NHIMs, or do they disappear at all The second possibility is that intersections between the stable and unstable manifolds of NHIMs change into tangency. This could lead to bifurcation in the way NHIMs are connected by their stable and unstable manifolds. 

We will also discuss the relevance, in the context of chemical reactions, of the possibilities pointed out in the third program stage, which enables us to go beyond the condition of normal hyperbolicity, and the need for the gap in the characteristic time scales. Moreover, bifurcation in the skeleton can offer a mechanism by which reaction processes evolve, thereby opening a new arena in the study of chemical reactions. 

The third stage of our strategy is discussed in Sections IX and X. Our discussion is speculative, since quantitative analysis is lacking at present. In Section IX, we point out that, in reaction dynamics, breakdown of normal hyperbolicity would also play an important role. Such cases would include phase transitions in systems with a finite number of degrees of freedom. In Section X, we will discuss the possibility of bifurcation in the skeleton of reaction paths, and we point out that it corresponds to crisis in multidimensional chaos. This approach offers an interesting mechanism for chemical evolution. 

Normal hyperbolicity enables us to derive the flow for e > 0 from the singular case with e = 0. This criterion was presented by Fenichel in Refs. 4-6 and was... 

Let Mq denote the manifold that is defined by the graph (xo(y),y) for s = 0. Roughly Speaking, normal hyperbolicity of the manifold Mq means that the absolute values of the Lyapunov exponents along the normal directions of Mq are much larger than those along the tangent directions of Mq. In other words, there exists a gap between them. 

As for the term invariant, it means that orbits starting on a NHIM stay on it at least locally in time for both forward and backward directions. However, due attention must be paid to the following. In general, a NHIM will have boundaries where orbits starting on it flow off the manifold. This is because the flow on it could reach those locations where normal hyperbolicity breaks down. Later in this chapter we will mention an example of this behavior. 

For the case of normal hyperbolicity, the theorem proved by Fenichel and independently by Hirsch et al. guarantees the following For a small and positive E, there exists a NHIM with stable and unstable manifolds, and VF , respectively. The NHIM varies smoothly with respect to the parameter . Moreover, and W also vary smoothly with respect to the parameter s at least locally near the NHIM Mg-. 

The condition that the matrix A has an inverse plays a crucial role in the above derivation. Here, we explain that this condition is satished when the manifold Mq is normally hyperbolic. Let us study the time development of a small deviation 5a from the manifold Mq under the fast time variable x. Substitute a (x) = Xo(y) + 6a (x) into Eq. (3) and note that the slow variable y is constant. Then, the linear equation for 8a is... 

Recall that normal hyperbolicity of Mq means that the movement along directions normal to Mq is hyperbolic even when we use the fast time variable. In other words, the matrix A has eigenvalues with nonzero real parts. Thus, A has an inverse. 

In this section, we consider the breakdown of the condition of normal hyperbolicity. First, we explain a simple example where breakdown of normal hyperbolicity leads to a bifurcation in reaction processes. In the Belousov-Zhabotinsky (BZ) reaction [40], the bifurcation from the stable fixed point to the limit cycle takes place through the breakdown of normal hyperbolicity. This is the simplest case where mathematical analyses are in progress [41]. 

Second, we point out the possibility that normal hyperbolicity breaks down for NHIMs with saddles as the energy of the vibrational modes increases at saddles. These cases seem to be much more difficult than that in the BZ reaction. At present, no attempt to analyze these cases has been made. However, considering that we face these cases frequently in reactions, the study of the breakdown of normal hyperbilicity is urgent. 

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