Synchronization of chaotic orbits: The influence of unstable periodic orbits

A chaotic trajectory can be synchronized with a desired unstable orbit (chaotic, periodic, or fixed point) by using a drive variable for which the response subsystem Lyapunov exponents (SLE's) are negative. Unexpectedly, for the Lorenz and R\"ossler systems, the SLE's obtained for synchronization with the fixed point showed good agreement with those obtained for chaotic orbits. For the Duffing oscillator, a similar agreement was found between the SLE's of the chaotic orbit and those of the unstable period-six orbit. It is conjectured that the SLE's of the chaotic orbit retain a memory of the periods of the orbit's origin.