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Title: Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback

Abstract

In this paper we discuss the control of complex spatio-temporal dynamics in a spatially extended nonlinear system (fluid model of Pierce diode) based on the concepts of controlling chaos in the systems with few degrees of freedom. A presented method is connected with stabilization of unstable homogeneous equilibrium state and the unstable spatio-temporal periodical states analogous to unstable periodic orbits of chaotic dynamics of the systems with few degrees of freedom. We show that this method is effective and allows to achieve desired regular dynamics chosen from a number of possible in the considered system.

Authors:
; ;  [1]
  1. Department of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, Saratov, 410012 (Russian Federation)
Publication Date:
OSTI Identifier:
20768754
Resource Type:
Journal Article
Resource Relation:
Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 16; Journal Issue: 1; Other Information: DOI: 10.1063/1.2168394; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BEAM-PLASMA SYSTEMS; CHAOS THEORY; CONTROL THEORY; DEGREES OF FREEDOM; ELECTRON BEAMS; EQUILIBRIUM; FEEDBACK; MICROWAVE RADIATION; NONLINEAR PROBLEMS; ORBITS; PERIODICITY; STABILIZATION

Citation Formats

Hramov, Alexander E., Koronovskii, Alexey A., and Rempen, Irene S. Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback. United States: N. p., 2006. Web. doi:10.1063/1.2168394.
Hramov, Alexander E., Koronovskii, Alexey A., & Rempen, Irene S. Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback. United States. doi:10.1063/1.2168394.
Hramov, Alexander E., Koronovskii, Alexey A., and Rempen, Irene S. Wed . "Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback". United States. doi:10.1063/1.2168394.
@article{osti_20768754,
title = {Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback},
author = {Hramov, Alexander E. and Koronovskii, Alexey A. and Rempen, Irene S.},
abstractNote = {In this paper we discuss the control of complex spatio-temporal dynamics in a spatially extended nonlinear system (fluid model of Pierce diode) based on the concepts of controlling chaos in the systems with few degrees of freedom. A presented method is connected with stabilization of unstable homogeneous equilibrium state and the unstable spatio-temporal periodical states analogous to unstable periodic orbits of chaotic dynamics of the systems with few degrees of freedom. We show that this method is effective and allows to achieve desired regular dynamics chosen from a number of possible in the considered system.},
doi = {10.1063/1.2168394},
journal = {Chaos (Woodbury, N. Y.)},
number = 1,
volume = 16,
place = {United States},
year = {Wed Mar 15 00:00:00 EST 2006},
month = {Wed Mar 15 00:00:00 EST 2006}
}
  • We show that the [ital continuous] feedback approach is highly effective for controlling chaotic systems. The control design for the Lorenz system is presented as an example to demonstrate the strength of this approach. The proposed control is able to eliminate chaos and bring the system toward any of the three steady states. Two different control input locations are considered. Only one system variable is used in the feedback. The control scheme can tolerate both measurement noise and modeling uncertainty as long as they are bounded.
  • The spectrum of Lyapunov exponents is powerful tool for the analysis of the complex system dynamics. In the general framework of nonlinear dynamics, a number of the numerical techniques have been developed to obtain the spectrum of Lyapunov exponents for the complex temporal behavior of the systems with a few degree of freedom. Unfortunately, these methods cannot be applied directly to analysis of complex spatio-temporal dynamics of plasma devices which are characterized by the infinite phase space, since they are the spatially extended active media. In the present paper, we propose the method for the calculation of the spectrum ofmore » the spatial Lyapunov exponents (SLEs) for the spatially extended beam-plasma systems. The calculation technique is applied to the analysis of chaotic spatio-temporal oscillations in three different beam-plasma model: (1) simple plasma Pierce diode, (2) coupled Pierce diodes, and (3) electron-wave system with backward electromagnetic wave. We find an excellent agreement between the system dynamics and the behavior of the spectrum of the spatial Lyapunov exponents. Along with the proposed method, the possible problems of SLEs calculation are also discussed. It is shown that for the wide class of the spatially extended systems, the set of quantities included in the system state for SLEs calculation can be reduced using the appropriate feature of the plasma systems.« less
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