Synchronization and control of directed transport in chaotic inertial ratchets via passive control

In this paper, transport control and synchronization are investigated between two periodically driven nonidentical, inertial ratchets that are able to exhibit directed transport. One of the two ratchets is acting as a drive system while the other one represents the response system. Based on the Lyapunov stability theorem, the essential conditions, under which the error nonlinear system is transformed into an equivalent passive system and globally asymptotically stabilized at equilibrium points, are established. With these results, synchronization not only between two nonidentical ratchets with known parameters but also between two different uncertain ratchets are realized via adaptive passive controllers and parameter update algorithm. The direction of transporting particles can be dominated along expected one and it is useful to control the motion of tiny particles, ratchetlike devices in nanoscience.