Controlling unstable periodic orbits by a delayed continuous feedback

The control of chaos: theory and applications

“…The experimental setup was an externally driven nonlinear oscillator with a tunnel diode as negative resistance device. Bielawski et al [167] use essentially the same diode resonator which was used in the experiment by Hunt [15], but with a higher frequency (about 10.3 MHz). The control signal is selected by comparing the output signal with the same signal delayed by a time corresponding to the period of the desired orbit.…”

On the stability of delayed feedback controllers

“…Hence asymptotic stability of Σ 1 for (1) and (4) can be analyzed by studying the stability of the corresponding fixed point of F for (5). To analyze the latter, let Σ 1 = {x * 0 } and set a 1 = Df (x * 0 ), and J = ∂F ∂x Σ 1 , where D stands for the derivative and J is the Jacobian of F evaluated at the equilibrium point.…”

“…DFC has been successfully applied to many systems, including the stabilization of coherent modes of laser [5,6]; cardiac systems, [7,8], controlling friction, [9]; chaotic electronic oscillators, [10,11]; chemical systems, [12]. To overcome the limitations of DFC, several modifications have been proposed, [13][14][15][16][17].…”

Stability of periodic orbits controlled by time-delay feedback

“…In general, U(τ ) must be obtained by direct numerical integration of Eqn. (10). In order for p(t) to be periodic, Eqn.…”

“…We note that versions of ETDAS that apply to discrete maps can be treated analytically and it is known that ETDAS can stabilize orbits that are uncontrollable using TDAS. [4,7] For continuous systems, both numerical results [3,8] and experiments [4,6,9,10] have shown that in order for ETDAS to be successful the feedback gain must lie within a finite, and often narrow, range. As the UPO is modified by changes in a bifurcation parameter this range of successful feedback gain will in general shift.…”