Tl. Carroll et Lm. Pecora, USING CHAOS TO KEEP PERIOD-MULTIPLIED SYSTEMS IN-PHASE, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 48(4), 1993, pp. 2426-2436
Periodically driven nonlinear systems can exhibit multiple-period beha
vior (period-2, period-3, etc.). Several such systems, when driven wit
h the same drive, can be on identical attractors but remain out of pha
se with each other (e.g., one drive cycle for a period-doubled set of
systems). This means that the basins of attraction for multiple-period
systems can be divided into domains of attraction, one for each phase
of the motion. A period-n attractor will have n domains of attraction
in its basin. This out-of-phase situation is stable-small perturbatio
ns will not succeed in getting the systems in phase. We show that one
can often use an almost periodic driving signal (generated from variou
s chaotic systems) which will simultaneously (1) keep the motion of th
e systems nearly the same as the periodic driving case, (2) keep the b
asin of attraction nearly the same, and (3) eliminate the n domains of
attraction. In other words, there will be only one domain for the bas
in. This means that any number of such driven systems will always be i
n phase. We display this effect in simulations and actual electrical c
ircuits, discuss the mechanism for this effect (which is most likely a
crisis), and speculate on some applications of the technique.