Onset of Chaos in the Generalized Ginzburg-Landau Equation

  • Boris A. Malomed
  • Alexander A. Nepomnyashchy
Part of the NATO ASI Series book series (NSSB, volume 225)


Study of chaos in the generalized Ginzburg-Landau equation (GLE)
$$ {\text{iu}}_{\text{t}} + {\text{u}}_{{\text{xx}}} + 2\left| {\text{u}} \right|^{\text{2}} {\text{u}} = {\text{i}} \in _1 {\text{u}} - {\text{i}} \in _3 \left| {\text{u}} \right|^2 {\text{u}} + {\text{i}} \in _2 u_{{\text{xx}}} $$
is a subject of great current interest, see, e.g., Refs. 1–6. We will consider Eq. (1) with the periodic boundary condition
$$ {\text{u(x)}} = {\text{u(x}} + 2\pi /{\text{k}}). $$


Modulational Instability Strange Attractor Lorenz System Instability Threshold Lorenz Model 
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  1. 1.
    Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from equilibrium, Progr. Theor. Phys. 55:356 (1976).ADSCrossRefGoogle Scholar
  2. 2.
    M. I. Rabinovich and A. L. Fabrikant, Stochastic self-modulation of waves in nonequi1ibrium media, Zh. Exp. Teor. Fiz. (Sov. Phys. — JETP) 77:617 (1979).ADSGoogle Scholar
  3. 3.
    H. T. Moon, P. Huerre, and L. G. Redekopp, Transition to chaos in the Ginzburg-Landau equation, Physica D 7:135 (1983).MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearity 1:279 (1960).MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    A. S. Pikovsky, Spatial development of chaos in nonlinear media, Phys. Lett. A 137:121 (1969).MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    G. M. Dudko and A. N. Slavin, Transition from modu-lational instability to chaos in films of iron-yttrium garnet (IYG), Fiz. Tv. Tela (Sov. Phys. — Sol. St. Phys.) 31:114 (1989).Google Scholar
  7. 7.
    E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20:130 (1963).ADSCrossRefGoogle Scholar
  8. 8.
    T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech. 27:417 (1967).ADSMATHCrossRefGoogle Scholar
  9. 9.
    T. Shimizu and N. Morioka, On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model, Phys. Lett. A 76:201 (1960).MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    C. Sparrow, “The Lorenz Equations: Bifurcations, Chaos and Strange Attractors”, Springer, New York (1962).Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Boris A. Malomed
    • 1
  • Alexander A. Nepomnyashchy
    • 2
  1. 1.P. P. Shirshov Institute for Oceanology of the USSR Academy of SciencesMoscowUSSR
  2. 2.Institute for Continuous Media Mechanics of the Ural Branch of the USSR Academy of SciencesPermUSSR

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