Mapping discrete

GrZa74] B. Griinbaum and J. Zaks, The existence of certain planar maps, Discrete Mathematics 10 (1974) 93-115. 

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.

We now examine how a next-amplitude-map was obtained from tire attractor shown in figure C3.6.4(a) [171. Consider tire plane in tliis space whose projection is tire dashed curve i.e. a plane ortliogonal to tire (X (tj + t)) plane. Then, for tire /ctli intersection of tire (continuous) trajectory witli tliis plane, tliere will be a data point X (ti + r), X (ti + 2r))on tire attractor tliat lies closest to tire intersection of tire continuous trajectory. A second discretization produces tire set Xt- = k = 1,2,., I This set is used in tire constmction... 

This section deals with the question of how to approximate the essential features of the flow for given energy E. Recall that the flow conserves energy, i.e., it maps the energy surface Pq E) = x e P H x) = E onto itself. In the language of statistical physics, we want to approximate the microcanonical ensemble. However, even for a symplectic discretization, the discrete flow / = (i/i ) does not conserve energy exactly, but only on... 

Fig. 4. Computation of the stochastic matrix Pa via mapping of discretization boxes.

Isoparametric mapping removes tlie geometrical inflexibility of rectangular elements and therefore they can be used to solve many types of practical problems. For example, the isoparametric C continuous rectangular Hermite element provides useful discretizations in the solution of viscoelastic flow problems. 

From a map at low resolution (5 A or higher) one can obtain the shape of the molecule and sometimes identify a-helical regions as rods of electron density. At medium resolution (around 3 A) it is usually possible to trace the path of the polypeptide chain and to fit a known amino acid sequence into the map. At this resolution it should be possible to distinguish the density of an alanine side chain from that of a leucine, whereas at 4 A resolution there is little side chain detail. Gross features of functionally important aspects of a structure usually can be deduced at 3 A resolution, including the identification of active-site residues. At 2 A resolution details are sufficiently well resolved in the map to decide between a leucine and an isoleucine side chain, and at 1 A resolution one sees atoms as discrete balls of density. However, the structures of only a few small proteins have been determined to such high resolution. 

Just as transient analysis of continuous systems may be undertaken in the. v-plane, stability and transient analysis on discrete systems may be conducted in the z-plane. It is possible to map from the. v to the z-plane using the relationship... 

Coupled-map Lattices. Another obvious generalization is to lift the restriction that sites can take on only one of a few discrete values. Coupled-map lattices are CA models in which continuity is restored to the state space. That is to say, the cell values are no longer constrained to take on only the values 0 and 1 as in the examples discussed above, but can now take on arbitrary real values. First introduced by Kaneko [kaneko83]-[kaneko93], such systems are simpler than partial differential equations but more complex than generic CA. Coupled-map lattices are discussed in chapter 8. 

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-... 

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

In many ways, May s sentiment echoes the basic philosophy behind the study of CA, the elementary versions of which, as we have seen, are among the simplest conceivable dynamical systems. There are indeed many parallels and similarities between the behaviors of discrete-time dissipative dynamical systems and generic irreversible CA, not the least of which is the ability of both to give rise to enormously complicated behavior in an attractive fashion. In the subsections below, we introduce a variety of concepts and terminology in the context of two prototypical discrete-time mapping systems the one-dimensional Logistic map, and the two-dimensional Henon map. 

Since the absolute value of the Jacobian J = a qn+i,Pn+i)/d qn,Pn) = 1, we see that this discrete-time map is indeed area-preserving. 

Consider, once again, a one-dimensional discrete-time map... 

The time evolution of the discrete-valued CA rule, F —> F, is thus converted into a two-dimensional continuous-valued discrete-time map, 3 xt,yt) —> (a y+i, /y+i). This continuous form clearly facilitates comparisons between the long-time behaviors of CA and their two-dimensional discrete mapping counter-... 

The case of multidimensional discrete-time mappings of the form... 

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

The axoneme consists of a cylinder of nine outer doublets of fused microtubules and a pair of discrete central microtubules (commonly referred to as the 9 + 2 arrangement of microtubules). The outer doublets each consist of a complete A-microtubule and an incomplete B-microtubule, the deficiency in the wall of the latter being made up by a sharing of wall material with the former. The tip of the axoneme contains the plus ends of all of the constituent microtubules. Two curved sidearms, composed of the MAP protein dynein, are attached at regular intervals to the A-microtubules of each fused outer doublet (Figures 1 and 2). 

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