Constraint-response plot

The behavior of complex dynamical systems can be analyzed and represented in a number of ways. Figure 1 represents one such approach, a constraint-response plot. A constraint, in this case [A], is any variable which the experimenter can control directly. A response, [X]ss in this case, is a measurable property of the system which depends upon the constraint values. The constraints are the external variables, e.g., the temperature of the bath surrounding the reactor or the reservoir concentrations, while the responses are the internal variables, e.g., the temperature or concentration of species in the reactor. The phase trajectory diagram of Fig. 4 is one type of response-response plot. Obviously, in a complex system, there will be several constraints and responses subject to independent (or coupled) variation. 

In order to really understand a system, we must study it under a variety of conditions, that is, for many different sets of control parameters. In this way, we will be able to observe whether bifurcations occur and to see how the responses of the system, such as steady-state concentrations or the period and amplitude of oscillations, vary with the parameters. Information of this type, which summarizes the results of a number of time series, is conveniently displayed in a constraint-response plot, in which a particular response, like a concentration, is plotted against a constraint parameter, like the flow rate. If the information is available, for example, from a calculation, unstable states can be plotted as well. Bifurcations appear at points where the solution changes character, and constraint-response plots are sometimes called bifurcation diagrams. An experimental example for a system that shows bistability between a steady and an oscillatory state is shown in Figure 2.13. 

Figure 2.13 This constraint-response plot indicates how the bromate-iodide reaction exists in steady states for high and low values of the residence time but oscillates for intermediate values. The distance between the dashed lines indicates the amplitude of the oscillations.

One advantage of the time series over the constraint-response plot or the phase diagram is that the time series tells us how our dynamical system evolves in time. Another way of looking at the evolution of a system is to view it in phase space. The phase portrait is similar to a time series in that we look at a fixed set of constraints (i.e., initial concentrations, flow rates, and temperature), but now we plot two or three dependent (concentration) variables to obtain a picture of the system s trajectory, or path through the phase space. A series of phase portraits at several sets of parameter values can provide a comprehensive picture of the system s dynamics. A calculated phase portrait for the Lotka-Volterra model and an experimental one for the BZ system are shown in Figure 2.15. 

Figure 13.7 A two-variable slow-fast system, (a) Constraint-response plot showing dependence of steady states of x on y, treated as a parameter (b) trajectory of oscillation in x-y plane (c) x oscillations (d) y oscillations. (Adapted from Rinzel, 1981.)...

The response surfaces in Figure 14.2 are plotted for a limited range of factor levels (0 < A < 10, 0 < B < 10), but can be extended toward more positive or more negative values. This is an example of an unconstrained response surface. Most response surfaces of interest to analytical chemists, however, are naturally constrained by the nature of the factors or the response or are constrained by practical limits set by the analyst. The response surface in Figure 14.1, for example, has a natural constraint on its factor since the smallest possible concentration for the analyte is zero. Furthermore, an upper limit exists because it is usually undesirable to extrapolate a calibration curve beyond the highest concentration standard. 

One analysis approach, appropriate if there are only a couple of design variables, is to construct contour plots of the mean response and the standard deviation of the response over the range of the variables. This will enable the researcher to see the constraints and trade-offs that may need to be made to achieve required values for the mean and variability of the response. 

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

Perhaps the most important feature of development chromatography is that the sample is separated by distance rather than time. This freedom from time constraints permits the utilisation of any of a variety of techniques to enhance the sensitivity of detection, such as reactions which increase light absorbance or fluorescence emission and wavelength selection for optimum response of each compound measured. The separation can be scanned as many times as desired, at a variety of wavelengths, and a complete UV visible or fluorescence spectrum can be easily plotted out for each component. Thus, the detection process in HPTLC is more flexible and variable than that for HPLC. Detection limits under optimum conditions are approximately the same for the two techniques. 

One may proceed in a similar fashion to obtain solutions for more and more tanks in series with equal time constants, with the constraint that the total time constant is equal to the sum of the identical individual time constants so that ti = T2 = = %n = %/n, creating systems of higher and higher order. These results are plotted in Figure 12.20 for the cases with the numbers of tanks in series n = 1,2,5,10, CO with M = 1 m min KpiKp2. .. Kp = 1 min m , t = 1 min. As the number of tanks increases, the response approaches that of a pure delay or dead time, 8, equal to the total time constant of the infinite tanks, 0 = z. 

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