Normal form theory

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

Most realistic Hamiltonians with simple saddles do not appear in this form. In what follows, we show how to transform such a Hamiltonian into this form using Normal-Form theory [13]. The phase-space structures that form the subject of this review will then be expressed in terms of the normal-form coordinates (qi,..., q ,pi,..., p ). Therefore, before analyzing Eq. (2) we show that any Hamiltonian vector field in the neighborhood of an equilibrium point of saddle (g) center (8> center type can be transformed to the form of Eq. (2). 

Finding a coordinate system that minimizes the coupling between the DOFs has always been a natural aspiration in theoretical chemistry. The so-called reaction-path formalism is just such a procedure, as is the use of Normal-Form theory [13], which is our method of choice. Normal-Form theory gives us sufficient conditions for a Hamiltonian to be transformed into the form of Eq. (1) in the neighborhood of an equilibrium point of center (g) center g) saddle type. This result is well known (see, e.g.. Ref. 13). To summarize, first we perform a Taylor expansion of the Hamiltonian [Eq. (1)] ... 

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. 

The phase space structures near equilibria of this type exist independently of a specific coordinate system. However, in order to carry out specific calculations we will need to be able to express these phase space structures in coordinates. This is where Poincare-Birkhoff normal form theory is used. This is a well-known theory and has been the subject of many review papers and books, see, e.g.. Refs. [34-AOj. For our purposes it provides an algorithm whereby the phase space structures described in the previous section can be realized for a particular system by means of the normal form transformation which involves making a nonlinear symplectic change of variables. 

To begin, let us see what all the several forms of Canonical Perturbation Theories (CPT) provide. All the CPTs [45-53], including normal form theories [54,55], require that an M-dimensional Hamiltonian H(p, q) in question be expandable as a series in powers of s, where the zeroth-order Hamiltonian is integrable as a function of the action variables J only... 

Canceling the Effect of Coupling Normal Form Theory... 

Instanton trajectories (solid) with several different energies calculated by the normal form theory are shown in the skewed coordinate system. The dotted curve shows the IRC. 

In Section 5.1, we have provided some evidence for the existence of saddle-node bifurcations, as a (local) mechanism accounting for the creation of pairs of solutions of our reaction-diffusion system (3) with Dirichlet boundary conditions. As the normal form theory [65, 66] shows, at least one of these two solutions is unstable. From a physical point of view, one can guess that, in the limit D oo, any stationary solution should be stable. However, as D is decreased, or equivalently, as the size of the system is increased, the system becomes approximately translationally invariant, and the stationary solutions are very likely to be unstable. As shown in this section, this transition involves not only (stationary) saddle-node bifurcations, but (oscillatory) Hopf bifurcations as well [62,104]. 

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