Devil staircase

Have you any experimental evidence of chaos and devil staircase in your system ... 

Thompson, A.H., Katz, A.J. and Raschke, R.A. (1987) Mercury injection in porous media a resistance devil s staircase with percolation geometry. Phys. Rev. Lett.,... 

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

Devil s staircase) Suppose that we pick a point at random from the Cantor set. What s the probability that this point lies to the left of x, where 0 [Pg.417]

Like other fractal concepts, the devil s staircase was long regarded as a mathematical curiosity. But recently it has arisen in physics, in connection with modelocking of nonlinear oscillators. See Bak (1986) for an entertaining introduction. 

W for K = 1 is shown in Figure 31. This figure is known as a devil s staircase because of its self-similar features. Notice that the blow-up of one small region of the staircase looks exactly like the entire staircase. An even higher resolution blow-up of a region between two small steps in the insert would, as well, resemble the entire staircase. This is the essence of self-similarity and is associated with the fact that the staircase has a fractal dimension. (The name devil s staircase is an indication of the frustration one would feel in trying to climb... 

Figure 31 The devil s staircase generated by solving the sine circle map for a range of values of (I with K = 1.0. (From Ref. 75 used with permission.)...

Thus, from the said above it follows, that thermooxidative degradation process of PAr and PAASO melts proceeds in the fractal space with dimension A In such space degradation process can be presented schematically as devil s staircase [33]. Its horizontal sections correspond to temporal intervals, where the reaction does not proceed. In this case the degradation process is described with fractal time t using, which belongs to Cantor s setpoints [34]. If the reaction is considered in Euclidean space, then time belongs to real numbers sets. 

Devalues, obtained for PAr and PUAr poly condensation process, showed, that the indicated processes were realized by aggre tion cluster-cluster mechanism [49], i.e., by small macromolecular coils joining in larger ones [23], Thus, polycondensation process is a fractal object with dimension D. reaction. Such reaction can be presented schematically in a form of devil s staircase [80], Its horizontal parts correspond to temporal intervals, in which reaction is not realized. In this case polycondensation process is described with irsing fractal time t, which belongs to Kantor s set points [81], If polycondensation process is considered in Euclidean space, then time belongs to a real number set. 

The Precursors Cantor s Set, the Devil s Staircase and the Peano-Hilbert Plane-Filling Curves... 

Figure 2J2 Intermediate stage (n = 6) in the construction of the Cantor singular function or Devil s staircase (top) and schematic illustration of the self-affinity of the Cantor singular function or DevU s staircase (bottom) the enlargement is identical to the original, but the enlargement (scaling) factors are different in the x and y directions.

In some crystals the location of dipole moments can even be more complicated. For example, in Fig. 13.15c, one layer with the dipoles looking down alternates with two layers where the dipoles are looking up. Therefore we have three-layer periodicity 3/ with two antiparallel layers and one extra polar layer. Such a structure may be considered as a mixture of the ferroelectric and antiferroelectric structures and is called ferrielectric. In case (c), the ferroelectric fraction is one part per period, qp = 1/3 and the spontaneous polarization is finite, Pg = (l/3)Po. For pure antiferroelectric phase qp = 0/2 and for pure ferroelectric one qp = 1/1 = 1. More generally, for different ferrielectric structures qp = nim, where m is the number of layers in the unit cell (period) and m is the ferroelectric layer fracture per unit cell, both being integers. Then, for both n and m oc, nim 1, the difference between n and m become smaller and smaller and the so-called Devil s staircase forms. 

Bminsma, R., Prost, J. Fluctuation forces and the Devil s staircase in ferroelectric smectic C s. J Phys. 11 France 4, 1209-1219 (1994)... 

Fig. 110. Mean-field phase diagram for the 3D Ising model with conq>eting interactions (Bak 1982). The dark areas indicate high-order C phases with I phases in-between (incomplete devil s staircase). P is the Lifshitz...

Hence, a solid-phase polymers deformation process is realized in fractal space with the dimension, which is equal to structure dimension d. In such space the deformation process can be presented schematically as the devil s staircase [39]. Its horizontal sections correspond to temporal intervals, where deformation is absent. In this case deformation process is described with using of fractal time t, which belongs to the points of Cantor s set [30]. If Euclidean object deformation is considered then time belongs to real numbers set. 

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