Delayed Feedback Control of Periodic Orbits in Autonomous Systems

Abstract

For controlling periodic orbits with delayed feedback methods the periodicity has to be known a priori. We propose a simple scheme, how to detect the period of orbits from properties of the control signal, at least if a periodic but nonvanishing signal is observed. We analytically derive a simple expression relating the delay, the control amplitude, and the unknown period. Thus, the latter can be computed from experimentally accessible quantities. Our findings are confirmed by numerical simulations and electronic circuit experiments.

Control of unstable steady states by extended time-delayed feedback

“…(12) and (17) also in the case of ϕ = 0. These calculations are lengthy and do not produce more insight and thus are omitted here.…”

Control of noise-induced oscillations by delayed feedback

“…Recently, the influence of the feedback as in [6] on a deterministically chaotic system was studied for a small range of τ in the vicinity of T 0 [24], and the ability of the feedback to change the period of a stabilized orbit was demonstrated.…”

Delayed feedback control of chaos: Bifurcation analysis

“…It has been shown numerically, however, that if τ is not equal, but close enough to T , the orbit changes both its shape and period. In [29] a method was proposed to find a better estimate for T from the knowledge of the periods of the stable periodic orbits at two different values of τ . The method works well if the two values of τ , which are chosen by an intial guess, are close enough to T , but its accuracy decreases if they are far from T .…”