Controlling the Chaotic Regime of Nonlinear Ionization Waves using the Time-Delay Autosynchronization Method

Delayed feedback control of unstable steady states with high-frequency modulation of the delay

“…(20). If ω were zero, the control term would have no preference for positive or negative values of ϕ because the change of ϕ → −ϕ would be equivalent to mirroring the complete system at one coordinate axis, e.g., x → −x.…”

“…The application is quite diverse, and includes stabilization of unstable states in electronic chaotic oscillators [5][6][7], mechanical pendulums [8,9], laser systems [10][11][12], electrochemical systems [13,14], drift waves in a magnetized * agjurcin@pmf.ukim.mk † schoell@physik.tu-berlin.de laboratory plasma [15], chaotic Taylor-Couette flow [16], cardiac systems [17,18], ferromagnetic resonance systems [19], gas discharge systems [20,21], controlling helicopter rotor blades [22], controlling the walking mechanism of a robot [23][24][25], stabilization of cantilever oscillations in an atomic force microscope [26], and controlling librational motion of a tethered satellite system in an elliptic orbit [27], amongst others. For a comprehensive review of the technical implementation of the method, see Refs.…”

Controlling Fast Chaos in Optoelectronic Delay Dynamical Systems

“…Note that ε(t) vanishes when the system is on the UPO since s(t) = s(t − τ) for all t. This control scheme has been successfully applied to diverse experimental systems such as electronic circuits [28,30,[41][42][43], Taylor-Couette fluid flow [44], an 15 NH 3 laser [45], a strongly driven magnetic system [46], plasma instabilities [47,48], and a chemical reaction [49,50]; see also other chapters of this Handbook. The simplicity of TDAS allows it to be implemented with much less latency than most control schemes.…”

An analytical treatment of the delayed feedback control algorithm at a subcritical Hopf bifurcation

“…By guessing only the period of the unstable orbit the system under control automatically settles on the desired periodic motion, and stability of this motion is maintained with only tiny perturbations. Successful implementation of this algorithm has been attained in quite diverse experimental systems including electronic chaotic oscillators [2][3][4][5], mechanical pendulums [6,7], lasers [8][9][10], gas discharge systems [11][12][13], a current-driven ion acoustic instability [14], a chaotic Taylor-Couette flow [15], chemical systems [16,17], high-power ferromagnetic resonance [18], helicopter rotor blades [19], and a cardiac system [20].…”