Oscillations in Three Dimensions

FIGURE 28 A three-dimensional limit-cycle and Poincare map. 

This case has been described in some detail to show how a comprehensive picture of one case can be presented. A gathering of similar presentations, a page for each of the eleven regions of the branch set, would provide a complete portrait of the Gray-Scott autocatalator. Systems of two equations with three parameters are open to such a complete treatment, but even a fourth would lead to a volume of such gatherings, whereas a fifth would demand a shelf, and a sixth, a library. A census of low-order dynamical systems to explore how far this might be taken would be an interesting exercise. 

The phase plane has to give place to the three-dimensional state-space when there are three dimensions. If we have a stable periodic solution such as is shown in Fig. 27, all trajectories in its neighborhood tend toward it, but this is harder to show in three dimensions than it is in two and showing that x(T + t) = x(t) when you do not know T is virtually impossible. Poincare solved this difficulty by setting up a plane transversal to the limit cycle, as in Fig. 28. The limit cycle penetrates this surface at P0 and, if a nearby trajectory penetrates it at a succession of points that, as shown, converge on P0, then 

Alhumaizi and R. Aris. Surveying a Dynamical System A Study of the Gray-Scott Reaction in a Two-Phase Reactor. Longman. Harlow (1995). 

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The quantityis dimensionless and is the ratio of the strength of the transition to that of an electric dipole transition between two states of an electron oscillating in three dimensions in a simple harmonic way, and its maximum value is usually 1. 

The energy of the isotropic harmonic oscillator in three dimensions can be written as... 

Now consider an ensemble of harmonic oscillators in three dimensions. Each of these harmonic oscillators has a different frequency oo = k c, their own Hamiltonian and raising and lowering operators... 

E. Schrodinger (1926), following the earher work of L. deBroglie (1924), advanced a fundamental equation of wave mechanics. The Schrodinger equation of wave mechanics was developed for waves oscillating in three dimensions with co-ordinates x, y and z ... 

In order to make use of equations such as (13 37) and (13 38) some assumption must be made with regard to the frequencies. As early as 1907 Einstein used the approximation of assuming that all of the 3 frequencies are equal. This is equivalent to supposing that each of the N atoms in the lattice makes quantized harmonic oscillations in three dimensions, and these oscillations are quite imaffected by the motion of the neighbouring atoms. This supposition of atomic independence cannot be correct, but nevertheless it leads to an... 

In order for these atoms to actually climb over the barrier from A to 6, they must of course be moving in the right direction. The number of times each zinc atom oscillates towards B is v/6 per second (there are six possible directions in which the zinc atoms can move in three dimensions, only one of which is from A to B). Thus the number of atoms that actually jump from A to B per second is... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

Another way of presenting time series of the type shown in Fig. 13 consists in deriving from them the attractors by applying the time delay method (74), i.e., by plotting A(f) versus A<1>(/ + r), whereby r is an arbitrary (but fixed) time, or—in three dimensions—by plotting A(() versus A(/ + 2r). Attractors of the latter type constructed from some of the time series of Fig. 13 are reproduced in Fig. 15 (71). Diagram a represents the simple periodic oscillations with the width... 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

In a series of articles, Rossler proposed abstract models which exhibit increasingly complicated oscillations. Most of these reaction schemes are in three dimensions and are accompanied by rate equations in discussing the behavior of the system. 

The oscillator strength was originally defined as unity for an electron oscillating harmonically in three dimensions, an early model of an atom. It can be determined experimentally by integration of an absorption band (Equation 2.20), where e is the molar absorption coefficient. 

Inequalities for oscillator strengths, previously used for estimating dipole polarizabilities in three dimensions, are generalized to D dimensions, and expressions for the dipole polarizability in the large D limit are obtained. The exact results, the dimensional scaling calculations, and the expressions obtained from inequalities are compared and evaluated. It is shown that the exact first order correction to the unperturbed wave function reduces to one term in the sum over states expression. The asymptotic result for the dipole polarizabilities is, in atomic units, 2 = (64Z ) D . 

The techniques described in this section are useful for studying chemical dynamics in the neighborhood of critical points. The remainder of this paper is devoted to the analysis of the global dynamics of nonlinear kinetic equations. In Section 2 a topological theorem is given, which can be used to place restrictions on the entire set of critical points of a chemical network. In Section 3 it is shown that many chemical networks can be classified on the basis of flows between volumes in concentration space. In Section 4, a number of techniques for establishing limit cycle oscillations in three and more dimensions are described. The topological methods are applied to analysis of compartmental chemical systems in Section 5. The results are discussed in Section 6. In the Appendix the principal mathematical results that have been used in the text are summarized. 

The simplest type of memory function approximation is to truncate the memory function hierarchy by assuming that the nth order memory function decays exponentially. This leads to a truncated continued fraction for the Fourier transform of Cj (t) with only one free parameter. Although the itinerant oscillator model in two dimensions is identical to a three term truncated continued fraction/ in three dimensions there is no such simple relationship this must be considered as an approximation with no prediction about C j j(t). 

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