Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T19:19:36.841Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  29 February 2012

Y. Kanevsky  and
A. A. Nepomnyashchy
Get access

Abstract

The influence of a global delayed feedback control which acts on a system governed by asubcritical complex Ginzburg-Landau equation is considered. The method based on avariational 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 theirlinear stability analysis is carried out. The application of the finite-dimensional modelallows to reveal the existence of chaotic oscillatory regimes and regimes withdouble-period and quadruple-period oscillations. The diagram of regimes resembles thosefound in the damped-driven nonlinear Schrödinger equation. The obtained results arecompared with the results of direct numerical simulations of the original problem.

Keywords

Ginzburg-Landau equationdelayed feedback controlfinite-dimensional modelssolitary waves
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 83 - 94
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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