Saddle-center equilibrium point

It is useful to note that the case of two DoF Hamiltonian systems is special. In this case a classical result of Ref. [75] (see also Ref. [76]) gives convergence results for the classical Hamiltonian normal form in the neighborhood of a saddle-center equilibrium point. Recently, the first results on convergence of the QNF have appeared. In Ref. [77] convergence resulfs for a one and a half DoF system (i.e., time-periodically forced one DoF Hamiltonian system) have been given. It is not unreasonable that these results can be extended to the QNF in the neighborhood of a saddle-center equilibrium point of a fwo DoF system. 

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)] ... 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

Phase Space. The PODS structure is easily lifted into phase space and described in a way very analogous to the linear case. We begin by finding the equilibrium points of the original Hamiltonian. Let P,- be such points. In a general case of chemical relevance, there will be a point P whose linear stability will be of stable/unstable (center/saddle) character. That is. 

Drawing Linear and Nonlinear NHIM. The equilibrium point Pi is thus of center/center/saddle type. The linearized motion in the vicinity lends itself to several pictures worth describing here. Let us begin with configuration space. The linearized potential energy may be written as... 

As mentioned in the previous section, reaction type dynamics is induced by equilibrium points of saddle x center x x center stability type. These are equilibria for which the matrix associated with the linearization of Hamilton s equations has eigenvalues which consist of a pair of real eigenvalues of equal magnitude and opposite sign, (- - A, — A,), A, e R, and d — 1) pairs of complex conjugate purely imaginary eigenvalues, ( + io, — io ), o e R, for k = 2,... 

The integrals provide a natural definition of the term "mode" that is appropriate in the context of reaction, and they are a consequence of the (local) integrability in a neighborhood of the equilibrium point of saddle-center-----center stability type. Moreover, the expression of the normal form Hamiltonian in terms of the integrals provides us a way to partition the "energy" between the different modes. We will provide examples of how this can be done in the following. 

The strength of the coupling is controlled by the parameter e in Eq. (66). The vector field generated by the corresponding classical Hamilton function has an equilibrium point at (q p2, q, pi, p2, p ) = 0. For e sufficiently small (for given values of parameters of the Eckart and Morse potentials), the equilibrium point is of saddle-center-center stability type. 

DF spectra until the optimized unzipping algorithm [11] has been carefully tested. Instead, SEP spectra of HCP will be discussed in which the cohx o>cp to bend resonance structure changes (HX is the H stretch against the CP center of mass) from 5 2 1 near HCP equilibrium to 3 2 1 near the HPC saddle point. 

The first order quantity determines positions of equilibrium or saddle points along the axis through the center ... 

The model contains two fixed points the trivial equilibrium R, N ) = (0, 0) where both predator and prey disappear and the feasible equilibrium (R, N ) = (6//c2, a/fci). Interestingly, in the feasible equilibrium the abundance of the predator or prey population is independent of their own respective growth or death rates but instead is set by the productivity of the other species. Stability analysis reveals that the trivial equilibrium is a saddle point and the feasible equilibrium is a neutrally stable center with... 

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