Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control

Parmananda, P.; Madrigal, Róger; Rivera, Marco; Nyikos, Lajos; Kiss, István Z.; Gáspár, Vilmos

doi:10.1103/physreve.59.5266

Cited by 105 publications

(64 citation statements)

References 22 publications

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“…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].…”

Section: Introductionmentioning

confidence: 99%

On the stability of delayed feedback controllers

Morgül

2003

Physics Letters A

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We consider the stability of delayed feedback control (DFC) scheme for one-dimensional discrete time systems. We first construct a map whose fixed points correspond to the periodic orbits of the uncontrolled system. Then the stability of the DFC is analyzed as the stability of the corresponding equilibrium point of the constructed map. For each periodic orbit, we construct a characteristic polynomial whose Schur stability corresponds to the stability of DFC. By using Schur-Cohn criterion, we can find bounds on the gain of DFC to ensure stability.

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Section: Introductionmentioning

confidence: 99%

On the stability of delayed feedback controllers

Morgül

2003

Physics Letters A

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“…Although time-delayed feedback control has been widely used with great success in real world problems in physics, chemistry, biology, and medicine, e.g. [6][7][8][9][10][11][12][13][14][15][16][17][18][19], a severe limitation used to be imposed by the common belief that certain orbits cannot be stabilized for any strength of the control force. In fact, it had been contended that periodic orbits with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the Pyragas method [20][21][22][23][24][25], even if the simple scheme (2.1) is extended by multiple delays in form of an infinite series [26].…”

Section: Z(t) = F (λ Z(t)) + B[z(t − τ) − Z(t)]mentioning

confidence: 99%

Delay Stabilization of Rotating Waves without Odd Number Limitation

Fiedler

Flunkert

Georgi

et al. 2008

Reviews of Nonlinear Dynamics and Complexity

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A variety of methods have been developed in nonlinear science to stabilize unstable periodic orbits (UPOs) and control chaos [1], following the seminal work by Ott, Grebogi and Yorke [2], who employed a tiny control force to stabilize UPOs embedded in a chaotic attractor [3,4]. A particularly simple and efficient scheme is time-delayed feedback as suggested by Pyragas [5], which uses the difference z(t − τ) − z(t) of a signal z at a time t and a delayed time t − τ. It is an attempt to stabilize periodic orbits of (minimal) period T by a feedback control which involves a time delay τ = nT, for suitable positive integer n. A linear feedback example iṡ(2.1) whereż(t) = f (λ, z(t)) describes a d-dimensional nonlinear dynamical system with bifurcation parameter λ and an unstable orbit of (minimal) period T. The constant feedback control matrix B is chosen suitably. Typical choices are multiples of the identity or of rotations, or matrices of low rank. More general nonlinear feedbacks are conceivable, of course. The main point, however, is that the Pyragas choice τ P

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“…The anodic voltage for this set of experiments was modulated as V Z V 0 C DxC gðI ðtÞKI ðtKtÞÞ, where gðI ðtÞKI ðtKtÞÞ was the feedback term intended to enhance the regularity of the spike trains. The control amplitude g and the delay time t were determined as follows (Parmananda et al 1999;Escalera Santos et al 2006). Using the time series provoked by the noise amplitude in figure 4b, a return map (not shown) was constructed by plotting successive interspike intervals (t pC1 versus t p ).…”

Section: Coherence Resonance Controlmentioning

confidence: 99%

Interaction of noise with excitable dynamics

Santos¹,

Rivera²,

Escalona³

et al. 2007

Phil. Trans. R. Soc. A.

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In this paper, the interaction of noise with excitable dynamics of a three-electrode electrochemical cell is examined. Different scenarios involving both external and internal noise sources are considered. In the case of external noise, aperiodic stochastic resonance and regulation of the noise-induced spiking behaviour are investigated. In the case of internal noise, the interaction of intrinsic electrochemical noise with autonomous nonlinear dynamics is studied. The amplitude of this internal noise, determined by the concentration of chloride ions, is monotonically increased and the provoked dynamics are analysed. Our results indicate that internal noise, similar to its external counterpart, is able to induce regularity in the system response.

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Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control

Cited by 105 publications

References 22 publications

On the stability of delayed feedback controllers

On the stability of delayed feedback controllers

Delay Stabilization of Rotating Waves without Odd Number Limitation

Interaction of noise with excitable dynamics

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