Dynamical control of the chaotic state of the current-driven ion acoustic instability in a laboratory plasma using delayed feedback

Abstract

Controlling chaos caused by the current-driven ion acoustic instability is attempted using the delayed continuous feedback method, i.e., the time-delay auto synchronization (TDAS) method introduced by Pyragas [Phys. Lett. A 170 (1992) 421.]. When the control is applied to the typical chaotic state, chaotic orbit changes to periodic one, maintaining the instability. The chaotic state is well controlled using the TDAS method.It is found that the control is achieved when a delay time is chosen near the unstable periodic orbit corresponding to the fundamental mode. Furthermore, when the delayed feedback is applied to a periodic nonlinear regime and arbitrary time delay is chosen, the periodic state is leaded to various motions including chaos. As a related topic, the synchronization between two instabilities of autonomous discharge tubes in a glow discharge is studied. Two tubes are settled independently and interacting each other through the coupler consisted of variable resister and capacitor. When the value of resister is changed as the strength of coupling, coupled system shows a state such as chaos synchronization.

Delay Stabilization of Rotating Waves without Odd Number Limitation

“…The mechanism of stabilization of the Pyragas orbit by a transcritical bifurcation relies upon the possible existence of such delay-induced periodic orbits with T = τ. Technically, the proof of the odd-number limitation theorem in [21] fails because the trivial Floquet multiplier μ = 1 (Goldstone mode of periodic orbit) was neglected there; F(1) in Eq. (14) in [21] is thus zero and not less than zero, as assumed [33]. At TC, where a second Floquet multiplier crosses the unit circle, this results in a Floquet multiplier μ = 1 of algebraic multiplicity two.…”

“…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].…”

Time-Delayed Feedback Control: From Simple Models to Lasers and Neural Systems

“…Although time-delayed feedback control has been widely used with great success in real world problems in physics, chemistry, biology, and medicine, e.g. [115,116,117,73,118,119,64,120,121,122,71,72,38], severe limitations are imposed by the common belief that certain orbits cannot be stabilized for any strength of the control force. In fact, it has been contended that periodic orbits with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the Pyragas method [43,44,123,124,125,126], even if the simple scheme is extended by multiple delays in form of an infinite series [39].…”

Stabilization of Long-Period Periodic Orbits Using Time-Delayed Feedback Control

“…Second, the only information required a priori is the period τ of the target UPO, rather than a detailed knowledge of the profile of the orbit, or even any knowledge of the form of the original ODEs, which may be useful in experimental setups. The method has been implemented successfully in a variety of laboratory situations [2,3,4,5,6,7,8], as well as analytically and numerically in spatially extended pattern-forming systems [9,10,11,12]; more examples can be found in a recent review by Pyragas [13].…”