Quasi random systems

Figure 3-4 Poincar6 section of a model flow (the sine flow). Areas of chaotic motion appear as a random cloud of points, whereas segregated regions (i.e., islands) appear as empty regions, if no particles were originally placed inside. Tracers particles that are placed inside islands appear as sets of closed curves or KAM surfaces in quasi-peiiodic systems, and as points in a periodic system.

Metamodel or response surface-based methods perhaps provide the best balance between computational intensity and information about the partial variances due to input parameter imcertainties. In many cases, the development of an accurate metamodel can be achieved using a far smaller sample size than that required by FAST or Sobol s basic method. The metamodel is then used for calculating global sensitivity indices. In common with the Sobol method, HDMR, for example, is based on the analysis of variance. Where higher-order terms (>2) in the HDMR expansion are weak, global sensitivity indices can be achieved using a relatively small quasi-random sample even for large parameter systems. 

A quantitative explanation of this behavior has been somewhat difficult to obtain, however. There have been several proposals to identify the experimentally resolved exponentials with physical entities, such as multiple monomer states [17-20] or multiple excimer states [15,21]. It is quite probable that the kinetics of the aryl vinyl pol)nners are indeed more complex than the simple Birks scheme would allow. However, recent theoretical studies on electronic excitation transport in random systems of donors and traps have shown that the fluorescence decays are in general, nonexponential [4,5, 22-26]. A key feature of these analyses is that the trapping dynamics in one dimensional and quasi one dimensional polymeric systems require that a trapping rate function k(t) rather than a trapping rate constant be used. In Section 2.1 we give the relationship of k(t) to the observables in a trapping experiment and provide the connection with G (t), which is obtained from theory. 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

The dynamic stability of the quasi steady-state process suggests that active control of the CZ system has to account only for random disturbances to the system about its set points and for the batchwise transient caused by the decreasing melt volume. Derby and Brown (150) implemented a simple proportional-integral (PI) controller that coupled the crystal radius to a set point temperature for the heater in an effort to control the dynamic CZ model with idealized radiation. Figure 20 shows the shapes of the crystal and melt predicted without control, with purely integral control, and with... 

So, apart from the regular behavior, which is either steady-state, periodic, or quasi-periodic behavior (trajectory on a torus, Figure 3.2), some dynamic systems exhibit chaotic behavior, i.e., trajectories follow complicated aperiodic patterns that resemble randomness. Necessary but not sufficient conditions in order for chaotic behavior to take place in a system described by differential equations are that it must have dimension at least 3, and it must contain nonlinear terms. However, a system of three nonlinear differential equations need not exhibit chaotic behavior. This kind of behavior may not take place at all, and when it does, it usually occurs only for a specific range of the system s control parameters 9. 

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