Structure and Dynamics of Spiral Waves and of Defects in Travelling Waves

  • E. Bodenschatz
  • A. Weber
  • L. Kramer
Part of the NATO ASI Series book series (NSSB, volume 244)


Systems with a bifurcation to an oscillatory homogeneous state (type 1) or to an oscillatory and spatially periodic state (type 2) appear in biology, chemistry and physics. The most prominent examples for type 1 are in biology the aggregation of social amoebae [2] and in chemistry the Belousov-Zhabotinskii (BZ) reaction [17]. Examples for type 2 are found in physical systems and are electro-convection of liquid crystals [19] and thermo-convection of binary-fluid mixtures [1]. These systems exhibit a Hopf bifurcation which may be super- or subcritical. Although in the case of the BZ reaction most experiments are performed in a parameter range, where the system behaves in an excitable manner, the description of the system as an oscillatory one captures many qualitative features.


Periodic Solution Hopf Bifurcation Spiral Wave Ginzburg Landau Equation Complex Ginzburg Landau Equation 
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  1. [1]
    See the contributions of Ahlers, G., Cannell, D.S. & Heinrichs, R.S. (1987). In Chaos 87, International Conference on the Physics of Chaos and Systems far from Equilibrium, Duong-Van, M. (ed.). North Holland: Amsterdam; and (1987) Nuclear Physics B (Proc. Suppl.) 2; Passner, A., Williams, H.L. & Surko, CM., ibid.; Steinberg, V., Moses, E. & Fineberg, J., ibid..Google Scholar
  2. [2]
    Alcantara, F. & Monk, M. (1974). Signal propagation during aggregation in the slime mould Dictyostelium discoideum. J. Gen. Microbiol. 85, 321.CrossRefGoogle Scholar
  3. [3]
    Bodenschatz, E. (1989). Pattern and Defects within the Weakly Nonlinear Analysis of Anisotropic Pattern-Forming Systems. Ph.D. Thesis, University of Bayreuth.Google Scholar
  4. [4]
    Bohr, T., Pedersen, A.W. & Jensen, M.H. (1989). Transition to Turbulence in a Discrete Ginzburg-Landau Model, Preprint.Google Scholar
  5. [5]
    Coullet, P. & Lega, J. (1988). Defect-Mediated Turbulence in Wave Patterns, Europhys. Lett.7, 511.ADSCrossRefGoogle Scholar
  6. [61]
    Coullet, P., Gil, L. & Lega, J. (1988). Defect mediated Turbulence, preprint.Google Scholar
  7. [7]
    Coullet, P., Gil, L. & Lega, J. A form of turbulence associated wirh defects, to appear in Physica D.Google Scholar
  8. [8]
    Elphick, C. & Meron, E. (1989). Spiral Vortex Interactions, Preprint.Google Scholar
  9. [9]
    Hagan, P.S. (1982). Spiral Waves in Reaction Diffusion Equations, SIAM J. Appl. Math. 42, 762.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Howard, L.N. & Kopell, N. (1977). Slowly Varying Waves and Shock Structures in Reaction-Diffusion Equations, Stud. Appl. Math. 56, 95.ADSzbMATHMathSciNetGoogle Scholar
  11. [11]
    Kopell, N. & Howard, L.N. (1973). Plane Wave Solutions to Reaction-Diffusion Equations, Stud. Appl. Math. 52, 291.zbMATHMathSciNetGoogle Scholar
  12. [12]
    Kramer, L., Bodenschatz, E., Kaiser, M., Pesch, W., Weber, A. and Zimmermann, W. (1989). Pattern and Defects in Liquid Crystals, in New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: the Geometry of Nonequilibrium, Coullet, P. & Huerre, P. (eds.). NATO ASI Series, Plenum Press.Google Scholar
  13. [13]
    Kuramoto, Y. (1984). Chemical Oscillations, Waves and Turbulence, Series in Synergetics. Springer: Berlin.CrossRefGoogle Scholar
  14. [14]
    Lega, J. & Gil, L.: private communication.Google Scholar
  15. [15]
    Malomed, B.A. (1983). Nonsteady Waves in distributed dynamical systems, Physic a 8D, 353.ADSMathSciNetGoogle Scholar
  16. [16]
    Malomed, B.A. (1984). Nonlinear Waves in Nonequilibrium Systems of the Oscillatory Type, Part I. Z. Phys. 55B, 241.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Müller, S.C., Plesser, T. & Hess, B. (1987). Two-dimensional Spectrophotometry of Spiral Wave Propagation in the Belousov- Zhabotinskii Reaction, Part I and II, Physica 24D, 71; (1987) Physica 24D, 87.Google Scholar
  18. [18]
    Newell, A.C. (1974). Envelope Equations. Lect. inAppl. Math. 15, 157.MathSciNetGoogle Scholar
  19. [19]
    See e.g. Rehberg, I., Rasenat, S. & Steinberg, V. (1989). Travelling Waves and Defect-Initiated Turbulence in Electroconvecting Nematics, Phys. Rev. Lett. 62, 756ADSCrossRefGoogle Scholar
  20. [20]
    Schöpf, W. & Zimmermann, W. (1990). Results om wave patterns in binary fluid convection, to appear in Phys. Rev. A41, 2.Google Scholar
  21. [21]
    Stewartson, K. & Stuart, J.R. (1971). A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. [21a]
    DiPrima, R.C., Eckhaus, W. & Segel, L.A.: Nonlinear wave-number interaction in near-critical two-dimensional flows. J. Fluid Mech. 49, 705.Google Scholar
  23. [22]
    Stuart, J.T. & DiPrima, R.C. (1978). The Eckhaus and Benjamin-Feir resonance mechanism, Proc. R. Soc. London A362, 27.ADSCrossRefGoogle Scholar
  24. [23]
    Weber, A. (1989). Two-dimensional Defect-Structures as Solutions of Ginzburg Landau Equations without Potential. Diploma Thesis, University of Bayreuth.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • E. Bodenschatz
    • 1
  • A. Weber
    • 1
  • L. Kramer
    • 1
  1. 1.Physikalisches InstitutUniversität BayreuthBayreuthGermany

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