Phase space systems normally hyperbolic invariant manifold

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]. 

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. 

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]. 

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

For systems of n degrees of freedom, take a normally hyperbolic invariant manifold with 2r normal directions in phase space. Note that for Hamiltonian systems, the dimension of the normal directions in phase space is alway even, because the eigenvalues of the variational equation (Jacobi equation) is symmetric around the value 0. Thus, the dimension of the normally hyperbolic invariant manifold is 2n — 2r and, for its stable and unstable manifolds, their dimensions are 2n — r, respectively. The dimension of their homoclinic intersection, if it exists, is 2n — 2r in the 2n-dimensional phase space. When we consider the intersection manifold on the equi-energy surface, its dimension on the surface is 2n — 2r — 1. Thus, the dimension d of the intersection on the Poincare section is d = 2n — 2r — 2. 

The symmetry properties of Hamiltonian systems imply that the stable and the unstable manifolds of the hyperbolic saddle point qo,po) have equal dimensions in the full 6D phase space (g,p. Pi, P2, Qi, Q2) G x R x R x T. The unperturbed system 0 = 7 = 0 has a 4D (R x T ) normally hyperbolic invariant manifold given by the union of the hyperbolic saddle points qo,Po) according to... 

Most of the 2D nonresonant invariant tori T(Pi,P2)) that persist are only slightly deformed on the perturbed normally hyperbolic locally invariant manifold and are KAM tori. In the phase space of the perturbed system 7 > 0 and a = 0, there are invariant tori that are densely filled with winding trajectories that are conditionally periodic with two independent frequencies conditionally-periodic motions of the perturbed system are smooth functions of the perturbation 7. A generahzation of the KAM theorem states that the KAM tori have both stable and unstable manifolds by the invariance of manifolds, b fn order to determine if chaos exists, two measurements are required in order to determine whether or not and VK (T.y) intersect transversely. 

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