Transport control in a deterministic ratchet system
Woo-Sik Son,1,* Jung-Wan Ryu,2 Dong-Uk Hwang,3 Soo-Young Lee,4 Young-Jai Park,5,† and Chil-Min Kim1,‡
National Creative Research Initiative Center for Quantum Chaos Applications, Sogang University, Seoul 121-742, Korea 2 Department of Physics, Pusan National University, Busan 609-735, Korea 3Department of Biomedical Engineering, University of Florida, Gainesville, Florida 32611-6131, USA 4 Department of Physics, Seoul National University, Seoul 151-747, Korea 5 Department of Physics, Sogang University, Seoul 121-742, Korea Received 16 December 2006; revised manuscript received 10 March 2008; published 19 June 2008 We study the control of transport properties in a deterministic inertia ratchetsystem via the extended delay feedback method. A chaotic current of a deterministic inertia ratchet system is controlled to a regular current by stabilizing unstable periodic orbits embedded in a chaotic attractor of the unperturbed system. By selecting an unstable periodic orbit, which has a desired transport property, and stabilizing it via the extended delay feedback method, we can controltransport properties of the deterministic inertia ratchet system. Also, we show that the extended delay feedback method can be utilized for separation of particles in the deterministic inertia ratchet system as a particle’s initial condition varies. DOI: 10.1103/PhysRevE.77.066213 PACS number s : 05.45.Gg, 05.40. a, 05.45.Pq, 05.60.Cd
The ratchet effect, i.e., a directionalmotion of a particle using unbiased ﬂuctuations, has attracted much attention in recent years 1,2 . An early motivation in this ﬁeld is to explain an underlying mechanism of molecular motors which transport molecules in the absence of appropriate potential and thermal gradients 3 . The leading works in Refs. 4 and 5 open studies on the Hamiltonian ratchets 6 and the inertia ratchets 7 , respectively.Later, the ratchet effect has been studied theoretically and experimentally in many different ﬁelds of science, e.g., asymmetric superconducting quantum interference devices 8 , quantum Brownian motion 9 , Josephson-junction arrays 10 , application for separation of particles 11 , quenched disordered systems 12 , etc. It has been known that two conditions should be met to obtain the ratchet effect1 . First, a system has to be in a nonequilibrium state by a correlated stochastic 13 or a deterministic perturbation 14 . Second, the breaking of the spatial inversion symmetry is required. In doing so, an asymmetric periodic potential, named the “ratchet potential,” is introduced. In particular, several works concerning the control of ratchet dynamics have been presented. The applying of a weaksubharmonic driving in a deterministic inertia ratchet system was used to enlarge the parameter ranges where regular currents are observed 15 , and the signal mixing of two driving forces was considered to control transport properties in an overdamped ratchet system 16 . Also, the effect of time-delayed feedback 17 has been studied in ratchet systems 18 . Moreover, the anticipated synchronizationwas observed in delay coupled inertia ratchet systems 19 and the stabilization of chaotic current to low-period orbits was presented, using time-delayed feedback methods, in the deterministic inertia ratchet system 20 .
On the other hand, starting with the work of Ott, Grebogi, and Yorke 21 , various methods forcontrolling chaotic dynamics have been developed 22 . Particularly, Pyragas proposed a simple and efﬁcient method, which utilizes a control signal with a difference between the present state of the system and the previous state delayed by the period of an unstable periodic orbit UPO 23 . This method, which is called the Pyragas method or delayed feedback control, is noninvasive in the sense that the...