Tracking unstable periodic orbits in chaotic systems via time delayed feedback control

A time-delayed feedback control (TDFC) system is by nature a rather special version of the familiar autoregressive moving-average (ARMA) control, or the canonical state-space control systems. Despite some of its inherent limitations, TDFC can be quite successful in many chaos control applications. To understand to what extent the TDFC method is useful, some analytic (sufficient) conditions for chaos control from the TDFC approach are derived in this paper, for both stabilization and tracking problems. A gradient descent based search algorithm is incorporated with the TDFC to estimate the time delay constant for tracking unstable periodic orbits. The established theoretical results and estimation method are further clarified via a case study of the typical chaotic Rossler system with computer simulations.