N-dimensional system

In the past two decades, a variety of semiclassical initial-value representations have been developed [105-111], which are equivalent within the semiclassical approximation (i.e., they solve the Schrodinger equation to first order in H), but differ in their accuracy and numerical performance. Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator [105, 108, 187, 245, 252-255], which for a general n-dimensional system can be written as... 

In a symmetrical n-dimensional system the convective term must decrease as s" 1. The time dependence, such that the solution has the form C = C(ii), turns out to be... 

In mathematical terms lumping can be described as the reduction of the n-dimensional system described in equation (4.1) to an n-dimensional lumped set. 

Some terminology the phase space for the general system (2) is the space with coordinates x, x . Because this space is n-dimensional, we will refer to (2) as an n-dimensional system or an nth-order system. Thus n represents the dimension of the phase space. 

Poincare maps are useful for studying swirling flows, such as the flow near a periodic orbit (or as we ll see later, the flow in some chaotic systems). Consider an n-dimensional system x = f(x). Let 5 be an n-l dimensional surface of section (Figure 8.7.1). 5 is required to be transverse to the flow, i.e., all trajectories starting on S flow through it, not parallel to it. 

First, there are actually n different Liapunov exponents for an n-dimensional system, defined as follows. Consider the evolution of an infinitesimal sphere of perturbed initial conditions. During its evolution, the sphere will become distorted into an infinitesimal ellipsoid. Let 5j(r), k = l,...,zz, denote the length of the th principal axis of the ellipsoid. Then 5j(r) 5 (0), where the Aj are the Liapunov exponents. For large t, the diameter of the ellipsoid is controlled by the most positive Aj. Thus our A is actually the largest Liapunov exponent. 

The simplex search algorithm [232-235] is a common pattern search method. A simplex is a geometrical figure that has one more vertex than the dimension of the space in which it operates. That is, n + 1 for n-dimensional systems (3N + 1 for Cartesian coordinates with N atoms) or three vertices (i.e., a triangle) in the two-dimensional case demonstrated in Figure 4.5. 

Theorem 2. Suppose the n-dimensional system x = f(x) has a stable non-constant... 

Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator,which for a general n-dimensional system can be written as... 

Let ATo(0 denote a linearly stable T-periodic solution of an n-dimensional system of ordinary differential equations (2.1.1), or... 

The theorems of Bendixon and Dulac are generalized for the case of n-dimensional systems with first integrals. It is shown as an illustration that the classical Michaelis-Menten reaction has no oscillatory solution. As another illustration we show that reactions with M components composed of M-1 atoms have no oscillatory solution either C6j. 

Consider a p-parameter family of n-dimensional systems described by... 

One more codimension-one boundary of stability of periodic trajectories which corresponds to the blue sky catastrophe [152]. It may occur in n-dimensional systems where n > 3. 

It follows from the above theorem that the study of (C.3.5) is reduced to the study of the n-dimensional system... 

The general Gaussian wavefunction for a N-dimensional system at time t with spatial coordinates q is... 

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