Poincare

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

Peierls R 1935 Quelques proprietes typiques des oorps solides Ann. Inst. Henri Poincare 5 177-222... 

Figure C3.6.1 (a) WR single-handed chaotic attractor for k 2 = 0.072. This attractor is projected onto tire (c, C2) plane. The maximum value reached by c (t) is 54.1 and tire minimum reached by <7 " 2.5. The vertical line, at Cj = 8.5 for < 1, shows the position of tire Poincare section of tire attractor used later, (b) A projection, onto tire cft- ),cft2)) plane, of tire chaotic attractor reconstmcted from tire set of delayed coordinates cft),cft ),c (t2), where t = t + and t2 = t -I- T2, for 0 < t < 00, and fixed delays = 137 and T2 = 200. Note tliat botli cft ) and cftf) reach a maximum of P and a minimum of <"T""so tliat tire tliree-dimensional reconstmcted attractor is...

C3.6.1(a )), from right to left. Suppose that at time the trajectory intersects this Poincare surface at a point (c (tg), C3 (Sq)), at time it makes its next or so-called first reium to the surface at point (c (tj), c 3 (t )). This process continues for times t, .. the difference being the period of the th first-return trajectory segment. The... 

Figure C3.6.2 (a) The (fi2,cf) Poincare surface of a section of the phase flow, taken at ej = 8.5 with cq < 0, for the WR chaotic attractor at k = 0.072. (b) The next-amplitude map constmcted from pairs of intersection coordinates. ..,(c2(n-l-l),C2(n-l-2),C2(n-l-l)),...j. The sequence of horizontal and vertical line segments, each touching the diagonal B and the map, comprise a discrete trajectory. The direction on the first four segments is indicated.

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

Laboratoire de Chimie Physique et Microbiologie pour VEnvironnement, Unite Mixte de Recherche UMR 7564, CNRS - Universite H. Poincare Nancy I, 405, rue de Vandoeuvre, F-54600 Villers-les-Nancy, France E-mail walcarius Icpme. cnrs-nancy.fr... 

Gesztesy, F Grosse, H., and Thaller, B., 1984, A rigorous approach to relativistic corrections of bound state energies for spin-1/2 particles , Ann. Inst. Poincare 40 159. 

Their data translated into a power output of lOOW/kg. Where was the power coming from Was this finally an illimitable source of energy, nature s own perpetual-motion machine It was here that Einstein, in a concise phrase, carried the argument to its limit. In his own derivation of = mcr in 1905, five years after Poincare s observation, he remarked that any body emitting radiation should lose weight. 

Miller, A. I. (1987). A precis of Edmund Wliittakefs The Relativity Theoiy of Poincare and Lorentz. Ajchivcs Internationales d Histoire de Science 37 93—103. 

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

Poincare maps of this form have the obvious advantage of being much simpler to study than their differential-equation counterparts, without sacrificing any of the essential behavioral properties. They may also be studied as generic systems to help abstract behaviors of more complicated systems. 

Dissipative systems whether described as continuous flows or Poincare maps are characterized by the presence of some sort of internal friction that tends to contract phase space volume elements. They are roughly analogous to irreversible CA systems. Contraction in phase space allows such systems to approach a subset of the phase space, C P, called an attractor, as t — oo. Although there is no universally accepted definition of an attractor, it is intuitively reasonable to demand that it satisfy the following three properties ([ruelle71], [eckmanSl]) ... 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

In these developments there were two distinct stages the first one, which dealt with the nearly linear oscillations, found its perfect algorithm in the theories of Poincare (both topological and analytical), which constitute the major part of this review here the discoveries were most striking as well as systematic. Numerous phenomena that had remained as riddles for many years, sometimes even for centuries, were systematically explained. We indicate in Part I of this chapter the qualitative aspect of this progress and in Part II, the quantitative one. 

On the other hand, in view of the assumed analyticity of the function, we can make use of the theorem of Poincare, namely that the solution x(t,pup2,fi) can be represented by a series arranged according to the ascending powers of the parameters, that is... 

Bendixon, negative criterion of, 333 Bendixon-Poincare Theorem, 333 Berezetski, V. B., 723... 

Figure 2. Schematic of typical data and consistent Poincare sections from the quasiperiodic regime of Rayleigh-B nard convection. The rotation number W (in arbitrary units) is plotted versus Rayleigh number R for two different values...

The function P can be computed from either an analytical or a numerical representation of the flow field. In such a way, a 3-D convection problem is essentially reduced to a mapping between two-dimensional Poincare sections. In order to analyze the growth of interfacial area in a spatially periodic mixer, the initial distri-... 

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