Return map

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

Fig. 9.5 Schematic representation of Type-I intermittency. The plot on the LHS shows a return map for the system as it appears just below and precisely at the critical parameter value Tc. The plot on the RHS shows the return map for r > r. Note how, for r > Vc, X = Xc appears to first attract" then repel trajectories.

One of the big advantages of using methods like the Return Map is that they require inputs from all the functions that will contribute to the project. Not only must R D provide the development data, but also manufacturing must be involved in fore-... 

It is at this point the revenue constraints placed on the project can be challenged, if they are thought to be limiting the eventual success of the project. Evidence will need to be generated to prove this point. The use of techniques like the Return Map can be used to justify any extra spending (Section D, 2.3.2). 

D-12) House, C. H., Price, R. L.,The Return Map, Harvard Business Review, January-Feb-ruary 1991,92... 

A thorough analysis of chaotic oscillations for the NH3/O2 reaction over Pt has been performed by Sheintuch and Schmidt (2//). Bifurcation diagrams were presented in great detail, as well as phase plane maps and Fourier spectra. Figure 18 shows a series of oscillation traces obtained for various oxygen concentrations. By extracting a next return map from trace g in Fig. 18, evidence for intermittency could be obtained. 

The Poincare map is also called the first-return map, because... 

Chaos can further be characterized by resorting to Poincare sections. By determining, for example, the value a of the substrate corresponding to the nth peak, /3 , of product Pi in the course of aperiodic oscillations, we may construct the one-dimensional return map giving a i as a function of a (Decroly, 1987a Decroly Goldbeter, 1987). The continuous character of the curve thus obtained (fig. 4.11) denotes the deterministic nature of the chaotic behaviour. 

To characterize the different behavioural modes of the model, it is useful to simplify its dynamics by further reducing the dimension of the system. This can be done by constructing a one-dimensional return map. A Poincare section previously described (see fig. 4.11) can be utilized to this end. When plotting the concentration a +i corresponding to the (n + l)th peak of )3 as a function of a , we obtain, for the asymptotic regime attained by numerical integration of eqns (4.1), a certain number of points, finite or infinite, depending on the more or less complex nature of the oscillatory behaviour. 

Fig. 4.23. Poincare sections obtained in system (4.1) for different values of parameter kg (in s" ) (a) 1.537 (b) 1.5 (c) 1.534 (d) 1.539 (e) 1.86. Situations (c) and (e) correspond, respectively, to figs. 4.18d and 4.21. The continuous curve in (a) corresponds to chaos the simple or complex pattern of bursting obtained in the other cases is indicated. The results are obtained by integration of eqns (4.1). The construction of the return map a +i =/[( )] is explained in fig. 4.24 and in the text (Decroly Goldbeter, 1987).

Fig. 4.24. Schematic construction of the return map a ,i=/ [a (jSM)] used in fig. 4.23, from the oscillations generated by the model with multiple regulation (Decroly, 1987a).

Fig. 4.25. Piecewise linear map constructed on the basis of the results of fig. 4.23 to account for complex periodic oscillations of the bursting type. The onedimensional return map x +j=/(x ) is defined by eqns (4.5) which contain the three parameters M, a and b. The arrowed trajectory corresponds to the simple pattern of bursting with three peaks per period Tr(3). Parameter values are a = 6,b = 5,M=ll (Decroly Goldbeter, 1987).

The piecewise linear map also allows us to comprehend how the variation of another parameter, such as b, can elicit the transition from complex periodic oscillations to simple periodic behaviour. For such a transition to occur, it suffices that the segment/2(x ) acquire a less negative slope, so that the fixed point x of the return map becomes stable. 

The return map defined by eqns (4.11), and represented in the inset to fig. 4.29, takes this characteristic into account. This nonlinear map allows us to unify the various modes of simple or complex oscillatory behaviour observed in the differential system (4.1), including chaos (Decroly, 1987a Decroly Goldbeter, 1987) ... 

Fig. 4.29. The nonlinear return map represented in the inset and defined by eqns (4.11) accounts for simple and complex patterns of bursting, as well as for chaos, as a function of parameter A. Iterations of the return map are performed for M = 20 and B = 1 (Decroly Goldbeter, 1987).

When the maximum of a peak of intracellular cAMP in fig. 6.11 is plotted as a function of the maximum of the preceding peak, we obtain a return map yielding )8 +i as a function of /3 . This one-dimensional... 

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