Lorenz section

Figure 12.3.5 shows a Poincare section of the attractor. We slice the attractor with a plane, thereby exposing its cross section. (In the same way, biologists examine complex three-dimensional structures by slicing them and preparing slides.) If we take a further one-dimensional slice or Lorenz section through the Poincare section, we find an infinite set of points separated by gaps of various sizes. 

Then we have the same electric impulse in both cases. This gives the same electric gauge vector potential, 4,2. However, the Lorenz gauge potentials are quite different. For the electric dipole in Section VI, both 42 and 2 are zero. For the toroidal antenna equivalent electric dipole in Section VII, while 2 is zero, 42 is non zero. How then are these two cases different Within the gauge condition... 

Note that as discussed in previous sections, under static conditions, these two antennas give no fields. In going between two static conditions, one can have the same fields at intermediate times, but a change in the electric impulse, this being related to a change in the Lorenz vector potential or to a nonzero time integral of the gradient of the Lorenz scalar potential. However, with no fields, the vector potential has zero curl, which in a QED sense is not measurable. 

Here 0 are parameters. Ed Lorenz (1963) derived this three-dimensional system from a drastically simpl ified model of convection rolls in the atmosphere. The same equations also arise in models of lasers and dynamos, and as we ll see in Section 9.1, they exactly describe the motion of a certain waterwheel (you might like to build one yourself). 

In this section we ll follow in Lorenz s footsteps. He took the analysis as far as possible using standard techniques, but at a certain stage he found himself confronted with what seemed like a paradox. One by one he had eliminated all the known possibilities for the long-term behavior of his system he showed that in a certain range of parameters, there could be no stable fixed points and no stable limit cycles, yet he also proved that all trajectories remain confined to a bounded region and are eventually attracted to a set of zero volume. What could that set be And how do the trajectories move on it As we ll see in the next section, that set is the strange attractor, and the motion on it is chaotic. 

Could it be that all trajectories are repelled out to infinity No we can prove that all trajectories eventually enter and remain in a certain large ellipsoid (Exercise 9.2.2). Could there be some stable limit cycles that we re unaware of Possibly, but Lorenz gave a persuasive argument that for r slightly greater than, any limit cycles would have to be unstable (see Section 9.4). 

Second, the Lorenz map may remind you of a Poincare map (Section 8.7). In both cases we re trying to simplify the analysis of a differential equation by reducing it to an iterated map of some kind. But there s an important distinction To construct a Poincare map for a three-dimensional flow, we compute a trajectory s successive intersections with a two-dimensional surface. The Poincare map takes a point on that surface, specified by two coordinates, and then tells us how those two coordinates change after the first return to the surface. The Lorenz map is different because it characterizes the trajectory by only one number, not two. This simpler approach works only if the attractor is very flat, i.e., close to two-dimensional, as the Lorenz attractor is. 

As tools for analyzing differential equations. We have already encountered maps in this role. For instance, Poincare maps allowed us to prove the existence of a periodic solution for the driven pendulum and Josephson junction (Section 8.5), and to analyze the stability of periodic solutions in general (Section 8.7). The Lorenz map (Section 9.4) provided strong evidence that the Lorenz attractor is truly strange, and is not just a long-period limit cycle. 

To compare these results to those obtained for one-dimensional maps, we use Lorenz s trick for obtaining a map from a flow (Section 9.4). For a given value of c, we record the successive local maxima of x(r) for a trajectory on the strange attractor. Then we plot x,, vs. x , where denotes the th local maximum. This Lorenz map for c = 5 is shown in Figure 10.6.7. The data points fall very nearly on a one-dimensional curve. Note the uncanny resemblance to the logistic map ... 

These same issues confronted scientists in the mid-1970s. At the time, the only known examples of strange attractors were the Lorenz attractor (1963) and some mathematical constructions of Smale (1967). Thus there was a need for other concrete examples, preferably as transparent as possible. These were supplied by Henon (1976) and Rdssler (1976), using the intuitive concepts of stretching and folding. These topics are discussed in Sections 12.1-12.3. The chapter concludes with experimental examples of strange attractors from chemistry and mechanics. In addition to their inherent interest, these examples illustrate the techniques of attractor reconstruction and Poincare sections, two standard methods for analyzing experimental data from chaotic systems. 

Area contraction is the analog of the volume contraction that we found for the Lorenz equations in Section 9.2. As in that case, it yields several conclusions. For instance, the attractor A for the baker s map must have zero area. Also, the baker s map cannot have any repelling fixed points, since such points would expand area elements in their neighborhood. 

Thus E(R) is indeed propagated with the same velocity as P(R). This establishes the interpretation of n as the refractive index of the medium. We will discuss Eq. IV.24 which represents a generalization of the Lorentz-Lorenz formula in Section IV.E. 

Among these causes only those mentioned in point 3 are within the scope of this book. We shall discuss them in some detail in Section 6.5 in connection with present climatic variations. Of the other factors we shall clarify here only point 6, which is probably less known than the others. If in our climatic model (see later) the random modification of initial conditions results in a nonzero probability that the climate remains unchanged, then the climate is called stable. If the system has only one single stable climate (and possibly many unstable ones) it is said to be transitive (see more detail in SMIC, 1971). Otherwise, i is said to be intransitive. According to Lorenz (1968) the climate of the Earth is intransitive. This means that climate can go from one stable state to another without the modification of external or initial conditions, that is, variations are due to the internal fluctuations of the system. 

This attractor is called the Lorenz attractor. The Lorenz attractor falls into a class of so-called strange attractors, corresponding to the quasi--stochastic behaviour of a system. Dynamics of this type has been found to occur in the systems of chemical kinetics equations (see Section 6.3.3). 

These strongly non-linear equations require a numerical solution. The random nature of the numerical solving of this problem helped Lorenz to discover the chaotic behavior of the solutions in the sense of a strong dependency with respect to the initial conditions as given by the Webster definition quoted in Section 1. Indeed, the computer used by Lorenz encountered - as was common at that time - a breakdown. In 1963, the size of internal memories did not allow... 

As discussed in section 9.2.1, the refractive index of a substance depends on the polarisabilities of its basic units. The Lorentz-Lorenz equation derived there as equation (9.9) links the refractive index n and the polari-sability a of the structural units in any isotropic medium. It ean be written in the slightly simpler form... 

The turbidity of a highly dilute latex sample will provide information about the number and/or size of the polymer particles. If the system is sufficiently dilute to preclude multiple scattering, the turbidity at various wavelengths may be related to the concentration and size of the polymer panicles by Lorenz-Mie theory (see Section 12.3.2). This has been done by Heller and co-workers [31,32]. Since the method involves only sample dilution followed by turbidity analysis by a UV-visible spectrophotometer, it is a natural choice for continuous, online use. 

A = specimen cross-sectional area F = Lorenz force H = magnetic field... 

Fig. 7 compares the experimentally measured (A and C) absorption Cahs,x and (B and D) scattering Csca,x cross-sections between 400 and 700 nm of monodisperse latex spheres 2.02 and 4.5 pm diameter with Lorenz—Mie theory predictions using the complex index of refraction of latex reported by Ma et al. (2003). Flere also, the good agreement between theoretical and experimental results successfully validated the experimental setup and the data analysis. Similar vaHdation has been performed with the same polydisperse polystyrene latex microspheres and randomly oriented and infinitely long glass fibers considered for validating the scattering phase function measurements, as illustrated in Fig. 6 (Berberoglu and Pilon, 2007).

Figure 7 Experimental measurement and Lorenz-Mie theory predictions of the average absorption Cabs,i and scattering Csca,i cross-sections between 400 and 700 nm of monodisperse polystyrene latex microspheres with diameters d equal to (A and B) 2.02 pm and (C and D) 4.5 pm, respectively (Kandilian, 2014).

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