# 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

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topKanevsky, 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

top- I.S. Aranson, L. Kramer. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys., 74 (2002), 99–143. Zbl1205.35299
- 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.
- 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.
- 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.
- I.V. Barashenkov, E.V. Zemlyanaya. Stable complexes of parametrically driven, damped nonlinear Schrödinger solitons. Phys. Rev. Lett., 83 (1999), 2568-2571. Zbl0932.35187
- 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.
- M. Bondila, I.V. Barashenkov, M.M. Bogdan. Topography of attractors of the parametrically driven nonlinear Schrödinger equation. Physica D, 87 (1995), 314-320. Zbl1194.35399
- 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.
- S.H. Davis. Theory of Solidification. Cambridge University Press, Cambridge, 2001. Zbl0991.76002
- A.A. Golovin, A.A. Nepomnyashchy. Feedback control of subcritical oscillatory instabilities. Phys. Rev. E, 73 (2006), 046212.
- 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. Zbl0228.76074
- 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.
- 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. Zbl1189.37078
- B.A. Malomed. Variational methods in nonlinear fiber optics and related fields. Progress in Optics, 43 (2002), 69–191.
- J.D. Moores. On the Ginzburg-Landau laxer mode-locking model with 5th order saturable absorber term. Opt. Commun., 96 (1993), 65–70.
- 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. Zbl1162.93349
- K. Nozaki, N. Bekki. Exact solutions of the generalized Ginzburg-Landau equation. J. Phys. Soc. Jpn., 53 (1984), 1581–1582.
- N.R. Pereira, L. Stenflo. Nonlinear Schrödinger equation including growth and damping. Phys. Fluids, 20 (1977), 1733–1734. Zbl0364.35012
- S. Popp, O. Stiller, E. Kuznetsov, L. Kramer. The cubic complex Ginzburg-Landau equation for a backward bifurcation. Physica D, 114 (1998), 81–107. Zbl0934.35176
- J.A. Powell, P.K. Jakobsen. Localized states in fluid convection and multiphoton lasers. Physica D, 64 (1993), 132–152. Zbl0772.35070
- 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.
- 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.
- W. Schöpf, W. Zimmermann. Convection in binary fluids - amplitude equations, codimension-2 bifurcation, and thermal fluctuations. Phys. Rev. E, 47 (1993), 1739–1764.
- 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.
- 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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