Synergetics pp 34-38 | Cite as

Bifurcation of a Continuum of Unstable Modes

  • K. Kirchgässner
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 2)


In this contribution we consider model equations of increasing complexity which exhibit bifurcation in a continuum of points. In spite of the wealth of solutions present there is - under certain simple geometrical conditions - a selection principle yielding only a finite number of stable solutions. The geometrical conditions mentioned guarantee essentially “supercritical” bifurcation for the continuum as an entity and thus reflect at this level the now classical result for simple eigenvalue bifurcation [2], [7].


Periodic Solution Singular Solution Small Solution Simple Eigenvalue Nonzero Solution 
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  1. 1.
    Aronson, P.G., Weinberger, H.F. (1975), Nonlinear Diffusion in Population Genetics, Construction, and Nerve Propagation, Proc. Tulane Program in Partial Differential Equations, Lecture Notes in Math. No. 446, Springer Verlag, BerlinGoogle Scholar
  2. 2.
    Crandall, M.G., Rabinowitz, P.H., (1971), Bifurcation from Simple Eigenvalues, J. Functional Analysis, 8, pp.321–340MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    DiPrima, R.C., Eckhaus, W. and Segel, L.A., (1971), Nonlinear Wave Number Interaction in Near-Critical Two-Dimensional Flows, J. Fluid Mech. 49, pp. 705–744ADSMATHCrossRefGoogle Scholar
  4. 4.
    Fife, P.C., (1977), On Modelling Pattern Formation by Activator- Inhibitor-Systems, MRC Technical Symmary Report 1724, Univ. of Wisconsin, Math. Res. Center, MadisonGoogle Scholar
  5. 5.
    Fife, P.C., (1977), Stationary Patterns for Reaction-Diffusion Equations, MRC Technical Summaty Report 1709, Univ. of Wisconsin, Math. Res. Center, MadisonGoogle Scholar
  6. 6.
    Kirchgässner, K., (1977), Preference in Pattern and Cellular Bifurcation in Fluid Dynamics, to appear in: Applications of Bifurcation Theory, Proc. Advanced Seminar, Academic PressGoogle Scholar
  7. 7.
    Sattinger, D.H., (1971), Stability of Bifurcating Solutions by Leray - Schauder Degree, Arch. Rat. Mech. Anal., 43, pp. 154–166MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1977

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  • K. Kirchgässner

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