Controlling chaotic dynamical systems using time delay coordinates

Analysis of Observed Chaotic Data

“…If one has constructed the dynamical space using time delay reconstruction, then the dependence of the map on time dependent parameters is no longer solely on the previous value of the parameter, but it may depend, at step n + 1, on p(n) as well as p(n -1) [ND92,RGOD92]. This simple extension of the basic result is true if the time between Poincare sections is ~t > (dE -1 )T.…”

“…If the time between sections is less, then dependence on earlier values of p( n) will enter [ND92]. These additional dependencies on earlier values of the time dependent parameter do not change the general outlook for this control by parameter variation, but they do necessitate small alterations in the rule for choosing 8p( n) to drive the map onto the stable manifold.…”

Stabilizing Unstable Periodic Orbits via Universal Input?Output Delayed-Feedback Control

“…Furthermore, the only measurable signal may be a scalar output signal, e.g., y(·). In this case, delay coordinates [20] can be employed to represent the evolution of the system and to extract the quantities necessary for its control. The delay coordinates used to reconstruct the chaotic attractors can be classified into two types, namely those which are based on the continuous output signal (flow) denoted by y(t), and those which are based on the discrete output signal (Poincaré map) denoted by y(k).…”

“…The method assumes that the exact dynamic equations of the chaotic system are unknown and that only one scalar output signal is measurable. The delay coordinates [20] are utilized to reconstruct the strange attractor. Initially, the approximate input-output difference equation around the desired UPO is estimated using the parameter identification technique.…”

Control of chaos in multimode solid state lasers by the use of small periodic perturbations

“…The feedback techniques are very powerful and have proven to be very efficient; they allow one to target the nonlinear dynamical system to the desired trajectory. The most used feedback techniques are the OGY control [16],the OPF technique [17], self-controlled feedback [18], and delay coordinates [19]. The [20], the parametric excitation of an experimentally adjustable parameter [21], and taming the chaotic (or neutrally stable) system by the means of external periodic perturbation [22,23].…”