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Many-hole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion

We consider n small regions (referred to as holes) on a chaotic attractor and study the average lifetime it takes for a randomly initiated trajectory to land in their union. It is shown that the escape route through one of the holes may be considerably reduced by other holes, depending on their positions. This effect, referred to as shadowing, can significantly prolong the average lifetime. We construct the many hole interactions and interpret as the amount of shadowing between the holes.The effective range of these interactions is associated with the largest Lyapunov exponent. The shadowing effect is shown to be very large when the holes are located on n points of an unstable periodic orbit. Considerable attention is paid to this case since it is of interest to the field of controlling chaos.

chaotic transients ; holes on chaotic attractor ; controlled periodic motion

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