The Control of Chaos - Theory and Applications by S. Boccaletti et al.

By S. Boccaletti et al.

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1) points leading to Y . We call Y , Y ,2, Y the secondary path of the root path 1. This L   L procedure can be repeated for initial conditions chosen on a uniform grid of size in . Of course, if a trajectory goes to an undesirable attractor, we simply disregard this trajectory in the tree-building process. Finally, with suitably chosen , a hierarchy of paths to A in can be built with, say, N root paths. On each root path i, there can be some secondary paths, and on each secondary 0 path there can be third-order paths, etc.

Fig. 8a shows a controlled trajectory (solid line) to the desirable L attractor A , where both the size of the controlling neighborhood and the maximal allowed > parameter perturbation p are set to be 10\. The trajectory would asymptote to the un
 desirable attractor without control, as shown by the dotted line in Fig. 8a. When control is applied 140 S. Boccaletti et al. / Physics Reports 329 (2000) 103}197 Fig. 8. From Ref. [85]. (a) A controlled trajectory to the desirable attractor A , where the size of the controlling > neighborhood and the maximum allowed parameter perturbation p are both 10\.

Arrow indicates the instant at   which control is switched on. Fig. 5. ¹"50, AT with 10% noise (a) and 20% noise (b). 2. Same stipulations and   parameters as in the caption of Fig. 4. Arrows indicate the instant at which control is switched on. to understand that the range of validity of such approximation is having small values, which implies to be su$ciently close to the UPO to be controlled so as linear properties are valid. In Fig. 4 we report the application of our method to Eq. (63).

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