Hyperbolic manifold

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. 

Vol. 1902 A. Isaev, Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds (2007)... 

The surface t v-1 is a particular example of what Wiggins has referred to as a hyperbolic manifold and what De Leon and Ling have termed a normally invariant hyperbolic manifoldd Hyperbolic manifolds are unstable and constitute the formal multidimensional generalization of unstable periodic orbits. Hyperbolic manifolds, like PODS, can be either repulsive or attractive. - If motion near a hyperbolic manifold falls away without recrossing it in configuration space, the hyperbolic manifold is said to be repulsive. On the other hand, it is often the case that motion near a hyperbolic manifold will cross it several times in configuration space as it falls away, and in this case it is said to be attractive. 

Matveev, S. V., and Fomenko, A. T. Isoenergy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in the increasing order of their complicacy and computation of volumes of closed hyperbolic manifolds. Usp. Mat. Nauk (1988) (in print). 

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. 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

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... 

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