Next-amplitude map

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.

Figure C3.6.4 Single-handed chaotic attractor and next-amplitude map reconstmcted from experimental data for tire BZ reaction, (a) The reconstmcted attractor projected onto tire + x)) plane (see tire text for a...

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... 

Figure 8.19 (a) Poincare section formed by the intersection of trajectories in the three-dimensional phase space with a plane normal to the page and containing the dashed line in Figure 8.18b, (b) next amplitude map constructed from the data in Figure 8.18b. (Adapted from Swinney, 1983.)...

We describe here an approach to controlling chaos in chemical systems pioneered by Showalter and collaborators (Peng et al., 1991). The algorithm does not require knowledge of the underlying differential equations, but rather works from an experimentally determined 1-D map—for example, the next amplitude map for the system. We consider the problem of stabilizing an unstable period-1 orbit, that is, an unstable fixed point in the 1- D map, of a chaotic system. We assume that we have measured the 1-D map and express that map as... 

Figure 8.21 Example of the control algorithm where the next amplitude map is obtained from successive maxima of a bromide-sensitive electrode in the BZ reaction. Inset shows original map (left) and shifted map (right) in the neighborhood of the unstable steady state A. When the flow rate is shifted by 0.2%, the point A is shifted down to a point that evolves to the steady state A. (Adapted from Petrov et al., 1993.)... 

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