Low-Dimensional Description of Pulses under the Action of Global Feedback Control

Y. Kanevsky; A. A. Nepomnyashchy

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 2, page 83-94
  • ISSN: 0973-5348

Abstract

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The influence of a global delayed feedback control which acts on a system governed by a subcritical complex Ginzburg-Landau equation is considered. The method based on a variational principle is applied for the derivation of low-dimensional evolution models. In the framework of those models, one-pulse and two-pulse solutions are found, and their linear stability analysis is carried out. The application of the finite-dimensional model allows to reveal the existence of chaotic oscillatory regimes and regimes with double-period and quadruple-period oscillations. The diagram of regimes resembles those found in the damped-driven nonlinear Schrödinger equation. The obtained results are compared with the results of direct numerical simulations of the original problem.

How to cite

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Kanevsky, Y., and Nepomnyashchy, A. A.. "Low-Dimensional Description of Pulses under the Action of Global Feedback Control." Mathematical Modelling of Natural Phenomena 7.2 (2012): 83-94. <http://eudml.org/doc/222236>.

@article{Kanevsky2012,
abstract = {The influence of a global delayed feedback control which acts on a system governed by a subcritical complex Ginzburg-Landau equation is considered. The method based on a variational principle is applied for the derivation of low-dimensional evolution models. In the framework of those models, one-pulse and two-pulse solutions are found, and their linear stability analysis is carried out. The application of the finite-dimensional model allows to reveal the existence of chaotic oscillatory regimes and regimes with double-period and quadruple-period oscillations. The diagram of regimes resembles those found in the damped-driven nonlinear Schrödinger equation. The obtained results are compared with the results of direct numerical simulations of the original problem.},
author = {Kanevsky, Y., Nepomnyashchy, A. A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {Ginzburg-Landau equation; delayed feedback control; finite-dimensional models; solitary waves},
language = {eng},
month = {2},
number = {2},
pages = {83-94},
publisher = {EDP Sciences},
title = {Low-Dimensional Description of Pulses under the Action of Global Feedback Control},
url = {http://eudml.org/doc/222236},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Kanevsky, Y.
AU - Nepomnyashchy, A. A.
TI - Low-Dimensional Description of Pulses under the Action of Global Feedback Control
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/2//
PB - EDP Sciences
VL - 7
IS - 2
SP - 83
EP - 94
AB - The influence of a global delayed feedback control which acts on a system governed by a subcritical complex Ginzburg-Landau equation is considered. The method based on a variational principle is applied for the derivation of low-dimensional evolution models. In the framework of those models, one-pulse and two-pulse solutions are found, and their linear stability analysis is carried out. The application of the finite-dimensional model allows to reveal the existence of chaotic oscillatory regimes and regimes with double-period and quadruple-period oscillations. The diagram of regimes resembles those found in the damped-driven nonlinear Schrödinger equation. The obtained results are compared with the results of direct numerical simulations of the original problem.
LA - eng
KW - Ginzburg-Landau equation; delayed feedback control; finite-dimensional models; solitary waves
UR - http://eudml.org/doc/222236
ER -

References

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  1. I.S. Aranson, L. Kramer. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys., 74 (2002), 99–143.  
  2. I.V. Barashenkov, M.M. Bogdan, V.I. Korobov. Stability diagram of the phase-locked solitons in the parametrically driven, damped nonlinear Schrödinger equation. Europhys. Lett., 15 (1991), 113-118.  
  3. I.V. Barashenkov, Yu.S. Smirnov. Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons. Phys. Rev. E, 54 (1996), 5707-5725.  
  4. I.V. Barashenkov, Yu.S. Smirnov, N.V. Alexeeva. Bifurcation to multisoliton complexes in the ac-driven, damped nonlinear Schrödinger equation. Phys. Rev. E, 57 (1998), 2350-2364.  
  5. I.V. Barashenkov, E.V. Zemlyanaya. Stable complexes of parametrically driven, damped nonlinear Schrödinger solitons. Phys. Rev. Lett., 83 (1999), 2568-2571.  
  6. I.V. Barashenkov, E.V. Zemlyanaya. Soliton complexity in the damped-driven nonlinear Schrödinger equation : Stationary to periodic to quasiperiodic complexes. Phys. Rev. E, 83 (2011), 056610.  
  7. M. Bondila, I.V. Barashenkov, M.M. Bogdan. Topography of attractors of the parametrically driven nonlinear Schrödinger equation. Physica D, 87 (1995), 314-320.  
  8. S. Chávez Cerda, S.B. Cavalcanti, J.M. Hickmann. A variational approach of nonlinear dissipative pulse propagation. Eur. Phys. J. D, 1 (1998), 313–316.  
  9. S.H. Davis. Theory of Solidification. Cambridge University Press, Cambridge, 2001.  
  10. A.A. Golovin, A.A. Nepomnyashchy. Feedback control of subcritical oscillatory instabilities. Phys. Rev. E, 73 (2006), 046212.  
  11. L.M. Hocking, K. Stewartson. On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. Roy. Soc. Lond. A, 326 (1972), 289–313.  
  12. Y. Kanevsky, A.A. Nepomnyashchy. Stability and nonlinear dynamics of solitary waves generated by subcritical oscillatory instability under the action of feedback control. Phys. Rev. E, 76 (2007), 066305.  
  13. Y. Kanevsky, A.A. Nepomnyashchy. Dynamics of solitary waves generated by subcritical instability under the action of delayed feedback control. Physica D, 239 (2010), 87-94.  
  14. B.A. Malomed. Variational methods in nonlinear fiber optics and related fields. Progress in Optics, 43 (2002), 69–191.  
  15. J.D. Moores. On the Ginzburg-Landau laxer mode-locking model with 5th order saturable absorber term. Opt. Commun., 96 (1993), 65–70.  
  16. A.A. Nepomnyashchy, A.A. Golovin, V. Gubareva, V. Panfilov. Global feedback control of a long-wave morphological instability. Physica D, 199 (2004), 61–81.  
  17. K. Nozaki, N. Bekki. Exact solutions of the generalized Ginzburg-Landau equation. J. Phys. Soc. Jpn., 53 (1984), 1581–1582.  
  18. N.R. Pereira, L. Stenflo. Nonlinear Schrödinger equation including growth and damping. Phys. Fluids, 20 (1977), 1733–1734.  
  19. S. Popp, O. Stiller, E. Kuznetsov, L. Kramer. The cubic complex Ginzburg-Landau equation for a backward bifurcation. Physica D, 114 (1998), 81–107.  
  20. J.A. Powell, P.K. Jakobsen. Localized states in fluid convection and multiphoton lasers. Physica D, 64 (1993), 132–152.  
  21. B.Y. Rubinstein, A.A. Nepomnyashchy, A.A. Golovin. Stability of localized solutions in a subcritically unstable pattern-forming system under a global delayed control. Phys. Rev. E, 75 (2007), 046213.  
  22. W. Schöpf, L. Kramer. Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation for a subcritical bifurcation. Phys. Rev. Lett., 66 (1991), 2316–2319.  
  23. W. Schöpf, W. Zimmermann. Convection in binary fluids - amplitude equations, codimension-2 bifurcation, and thermal fluctuations. Phys. Rev. E, 47 (1993), 1739–1764.  
  24. V. Skarka, N.B. Aleksić. Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations. Phys. Rev. Lett., 96 (2006), 013903.  
  25. E.N. Tsoy, A. Ankiewicz, N. Akhmediev. Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation. Phys. Rev. E, 73 (2006), 036621.  

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