Abstract
Besides the occurrence of chaos in a large variety of natural processes, chaos may also occur because one may wish to design a physical, biological, or chemical experiment, or to project an industrial plant to behave in a chaotic manner. That chaos may indeed be desirable is further evidenced by the fact that it can be controlled using small perturbation of some accessible parameter or dynamical variable of the system.
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Notes
- 1.
For a larger target region, the exponent γ depends strongly on the location of I even if μ(I) is kept constant [101, 103, 571, 572, 574]. From the general theory of leaked systems (cf. Sect. 2.7), this can be understood as being due to the complicated overlap of the leak with its preimages.
- 2.
If the perturbation is chosen so that the trajectory falls on the other side, one can speed up the escape process to the simple attractor and can reduce the average lifetime of chaos [383].
- 3.
We address initial conditions only in the original basin of the attractor because, before the collapse, the system performs normally and operates in the precrisis regime. We are not concerned with initial conditions outside the basin, although they usually yield trajectories leading to V = 0. A voltage collapse can thus be regarded as a catastrophic event. Our control method is applicable to preventing this type of catastrophe.
- 4.
The stability of transient chaos against noise has been discussed in Chap. 4.
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© 2011 Springer Science+Business Media, LLC
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Lai, YC., Tél, T. (2011). Controlling Transient Chaos and Applications. In: Transient Chaos. Applied Mathematical Sciences, vol 173. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6987-3_11
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DOI: https://doi.org/10.1007/978-1-4419-6987-3_11
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