Time mapping

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-... 

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... 

Figure 8. Example of microwave conductivity transient map PMC relaxation time map taken from a 20- m thin silicon wafer onto which 11 droplets of zeolith suspension were deposited and dried. Reduced lifetimes are clearly observed in the region of droplets. For color version please see color plates opposite this page.

The response from the water and hydrocarbon can be distinguished by measuring the distributions of the diffusion coefficients simultaneously with the distributions of the relaxation times. The resulting distributions are displayed on a two-dimensional diffusion coefficient-relaxation time map and the distributions for... 

The spin-lattice relaxation time map (discussed in Section II.A.2) yields information about the spatial distribution of mean pore size within a given image pixel. Lighter shades in the image correspond to larger mean pore size. Even at this coarse... 

In an investigation of the spin-density (voidage) and spin-lattice relaxation time maps of many pellets, it was found that it was the heterogeneity in pore size, as characterized by the fractal dimension of the Ti map, that was consistent between images of pellets drawn from the same batch 58). The fractal dimensional of these images identifies a constant perimeter-area relationship for clusters of pixels of... 

To carry out this operation, a personal computer has proven indispensable. A group of programs implementing these ideas has been written by our colleague Peter Meyer. We call this program Timewave Zero. This software takes these theories and discoveries concerning the / Ching and creates time maps based upon them. These time maps or novelty maps show the ebb and flow of cormectedness or novelty in any span of time from a few days to tens of milletmia. The theory is not deterministic it... 

Lapping critical water quality indices in real time is necessary for the development and verification of realistic mathematical models of the ocean. The advent of automated chemical analyses and computer mapping (1-4) has made real-time mapping of static chemical properties a reality. But these static properties, for example, nutrient salts, chlorophyll, and salinity, are not sufficient to describe the state of a system, nor can they be used to predict the recovery of a perturbed system. The dynamic properties, especially those that control the remineralization and oxidation of organic matter to CO2 and NO3, namely oxygen consumption, denitrification, and nitrification, must be measured (5-8). These processes are... 

The predictive power of Feigenbaum s theory may strike you as mysterious. How can the theory work, given that it includes none of the physics of real systems like convecting fluids or electronic circuits And real systems often have tremendously many degrees of freedom—how can all that complexity be captured by a one-dimensional map Finally, real systems evolve in continuous time, so how can a theory based on discrete-time maps work so well ... 

Here mass transport is of minor importance compared to the changes in relaxation times induced by phase changes such as crystallization of water or lipid, phase separation, gelation and/or macromolecule aggregation and denatur-ation. Since relaxation times are sensitive to temperature, relaxation time maps might also be used to follow temperature changes and heat transport. 

We first reformulate unimolecular decay in terms of symbolic dynamics so as to permit utilization of modern concepts in ergodic theory. In doing so we, at least initially, replace the continuous time dynamics by a discrete time mapping. Specifically, we consider dynamics at multiples of a fixed time increment S, defining T"x as the propagation of a phase space point x for n time increments [i.e., x(t = nS) = T"x]. In what follows, time parameters associated with the discrete dynamics are measured in units of S. These include t, t, t, and t<, which are also used in connection with the flow. In the later context the conversion to -independent units is implicit. Note that within the assumed discrete dynamics, S and S+ are broadened from surfaces to volumes S-s and S+s comprising all points that enter or have left R during a time interval b. 

The results are discussed with relation to the relaxation time map. The distribution of Tg in a thin film is discussed in Sect. 6 for multilayered thin films utilizing isotope labeling by neutron reflectivity. Concluding remarks are given in Sect. 7. 

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