Analytical treatment of delayed feedback control

Synergetic System Analysis for the Delay-Induced Hopf Bifurcation in the Wright Equation

“…For instance, it may be interesting to revisit a Hopf bifurcation of codimension two in delayed systems as it occurs in some robotics applications [28]. Furthermore, the Floquet theory for delay differential equations, and thus the linear stability analysis for a periodic reference state, was already established in [29,30,31]. However, a corresponding synergetic system analysis which derives the order parameter equations and the normal forms for bifurcations of oscillatory solutions is still missing [10,11,32].…”

Analytical approach for the Floquet theory of delay differential equations

“…Finally the Floquet theory was completed by studying the adjoint problem. The applicability of our Floquet theory was demonstrated in [30]. In particular our analytical treatment provides means of understanding the mechanism of the continuous control of chaos by self controlling feedback [27,28].…”

“…In particular our analytical treatment provides means of understanding the mechanism of the continuous control of chaos by self controlling feedback [27,28]. Previous investigations have indicated that it becomes crucial to decide whether an observed stabilized limit cycle corresponds to an unstable cycle of the system or is produced by the control mechanism itself [29,30].…”

Control and Identification of Chaotic Systems by Altering the Oscillation Energy

“…The attractiveness of delayed feedback scheme consists in the self-organizing ability of a system to autosynchronize its own behaviour. However, unlike the OGY-based schemes where the trajectory is targeted to a predefined UPO, the delayed feedback control does not discriminate between different periodic orbits of the same period, and does not necessarily lead to the stabilization of orbits embedded in a chaotic attractor (Hikihara et al, 1997;Simmendinger et al, 1998). The success of this control is significantly restricted by a control loop latency (Just et al, 1999a).…”