Baker s map

A computer program (TWOPHASE) was developed that uses the Lockliart-Matinelli correlation and determines the total pressure drop based on the vapor phase pressure drop. The total length of the unit depends on the nature of the Reynolds numher. The program also calculates the gas-liquid phase regime employing a modified Baker s map [33]. Table 7-13 gives the results of the two-phase pressure drop. 

To conclude this section we discuss the baker s map (Farmer et al. (1983)) as an example for an area preserving mapping in two dimensions. Area preservation is of utmost importance for Hamiltonian systems, since Liouville s theorem (Landau and Lifechitz (1970), Goldstein (1976)) guarantees the preservation of phase-space volume in the course of the time evolution of a Hamiltonian system. The baker s map is a transformation of the unit square onto itself. It is constructed in the following four steps illustrated in Fig. 2.5. 

That this method is indeed quite efficient is demonstrated in the sequence of four frames shown in Fig. 2.6. Fig. 2.6(a) shows an orderly swirl of 50 raisins at the centre of the unit square. Fig. 2.6(b) shows the positions of the raisins after one application of the mapping (2.2.49). Fig. 2.6(c) shows the result after five applications. Although there are still noticeable correlations in the positions of the raisins, one can already see how the raisins tend to a homogeneous distribution. A very satisfactory result is achieved after only ten applications of the mapping, as shown in Fig. 2.6(d). The secret of this quick equilibration is the chaoticity of the baker s map. The baker s map is an example of how we can apply our knowledge about the shift map. A careful examination of... 

Because we want to reserve the word horseshoe for Smale s mapping, we have used the name pastry map for the mapping above. A better name would be the baker s map but that name is already taken by the map in the following example. 

The baker s map exhibits sensitive dependence on initial conditions, thanks to the stretching in the x-direction. It has many chaotic orbits—uncountably many, in fact. These and other dynamical properties ofthe baker s map are discussed in the exercises. 

The next example shows that, like the pastry map, the baker s map has a strange... 

Show that for a < +, the baker s map has a fractal attractor A that attracts all orbits. More precisely, show that there is a set A such that for any initial condition (. o > o) > the distance from 6"(Xq,> o) to A converges to zero as n —>. ... 

Find the box dimension of the attractor for the baker s map with a < 7. Solution The attractor A is approximated by which consists of 2"... 

For a < 7, the baker s map shrinks areas in phase space. Given any region R in the square,... 

This result follows from elementary geometry. The baker s map elongates R by a factor of 2 and flattens it by a factor of a, so area(5(7 )) = 2tzxarea(/ ). Since a < 7 by assumption, area(fi(7 )) < area(7 ) as required. (Note that the cutting operation does not change the region s area.)... 

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. 

In contrast, when a = 7 the baker s map is area-preserving. area(/ ( )) = area(7 ). Now the square 5 is mapped onto itself, with no gaps be-... 

In theculinary spirit ofthe pastry map and the baker s map, Otto Rossler (1976) found inspiration in a taffy-pulling machine. By pondering its action, he was led to a system of three differential equations with a simpler strange attractor than Lorenz s. The Rossler system has only one quadratic nonlinearity xz ... 

Sketch the face of Figure 12.1.4 after one more iteration of the baker s map. 

Vertical gaps) Let B be the baker s map with a < y. Figure 12.1.5 shows that the set S) consists of horizontal strips separated by vertical gaps of different sizes. 

Area-preserving baker s map) Consider the dynamics of the baker s map in the area-preserving case 0 = 7. 

Study the baker s map on a computer for the case 0 = 7. Starting from a random initial condition, plot the first ten iterates and label them. 

Figure 6.5 Sketch of the area preserving baker s map. The square is squeezed in the horizontal and stretched in the vertical direction, then it is cut in half to cover the original unit square.

Figure 3-2 Two components of chaotic mixing, stretching and folding, illustrated by a simple model. Baker s map defines a mixing protocol that stretches fiuid elements to double length and folds them in each unit time n. The amount of intermaterial contact area (the interface between the light and the dark regions) grows at an exponential rate as the recipe is applied repeatedly.

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