Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation

Delayed feedback control based on the act-and-wait concept

“…System (16) is stable when σ < 0, and system (17) is identical to system (9). This analytical argument is also valid for the other fixed point x * + .…”

“…Stability analysis of DFC systems is important for designing control systems; however, it is not easy because it requires solving the time-periodic linear systems, which include a time delay. In recent years, the stability of DFC systems has been intensively analyzed [5][6][7][8][9][10].…”

“…The stabilization of UFPs has been investigated analytically [9][10][11][12][13][14][15][16] and used in control theory [17] and laser systems [18]. Kokame et al derived a necessary and sufficient condition for the existence of a delayed feedback controller and provided a systematic procedure for its design [19].…”

An adaptive delayed feedback control method for stabilizing chaotic time-delayed systems

“…6 The stabilization of a fixed point of the logistic system by applying the DFC method with k = 2 when the variable system parameter is p = 6, p = 8, and p = 10 [41], and it has been shown that this model can exhibit chaos in a range of p. We consider that the initial condition is given by x 0 (θ ) = 0.5 where θ ∈ [−τ, 0]. The origin is an unstable equilibrium of (13) for any values of k [39]. But this model has one fixed point other than the origin at x * = 1 − 1/p which can be stabilized.…”

“…Several techniques have been proposed to adjust the delay for stabilizing UPOs [32][33][34][35][36][37]. For the case of chaotic timedelayed systems, there are also some analytical results in the recent literature to adjust the control parameters [38][39][40]. However, the DFC method is valid for chaotic systems only when the parameters of the system are fixed.…”