Normally hyperbolic invariant manifold NHIM

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]. 

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... 

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. 

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]. 

In statistical reaction rate theory, the concept of transition state plays a key role. Transition states are supposed to be the boundaries between reactants and products. However, the precise formulation of the transition state as a dividing surface is only possible when we consider transition states in phase space. This is the place where the concepts of normally hyperbolic invariant manifolds (NHIMs) and their stable and unstable manifolds come into play. 

Phase-space structure of Hamiltonian systems with multiple degrees of freedom—in particular, normally hyperbolic invariant manifolds (NHIMs), intersections between their stable and unstable manifolds, and the Arnold web. 

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