USING CHAOS TO KEEP PERIOD-MULTIPLIED SYSTEMS IN-PHASE

Periodically driven nonlinear systems can exhibit multiple-period behavior (period-2, period-3, etc.). Several such systems, when driven with the same drive, can be on identical attractors but remain out of phase 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 perturbations will not succeed in getting the systems in phase. We show that one can often use an almost periodic driving signal (generated from various chaotic systems) which will simultaneously (1) keep the motion of the systems nearly the same as the periodic driving case, (2) keep the basin of attraction nearly the same, and (3) eliminate the n domains of attraction. In other words, there will be only one domain for the basin. This means that any number of such driven systems will always be in phase. We display this effect in simulations and actual electrical circuits, discuss the mechanism for this effect (which is most likely a crisis), and speculate on some applications of the technique.