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Scaling Laws and Bifurcation

  • P. J. Aston
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1463)

Abstract

Equations with symmetry often have solution branches which are related by a simple rescaling. This property can be expressed in terms of a scaling law which is similar to the equivariance condition except that it also involves the parameters of the problem. We derive a natural context for the existence of such scaling laws based on the symmetry of the problem and show how bifurcation points can also be related by a scaling. This leads in some cases, to a proof of existence of bifurcating branches at a mode interaction. The results are illustrated for the Kuramoto-Sivashinsky equation.

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References

  1. Aston, P. J., Spence, A. and Wu, W. (1990). Bifurcation to rotating waves in equations with O(2)-symmetry. Submitted to SIAM J. Appl. Math. Google Scholar
  2. Barut, A. O. and Raczka, R. (1986). Theory of Group Representations and Applications, 2nd rev. ed. World Scientific, Singapore.Google Scholar
  3. Bröcker, T. and tom Dieck, T. (1985). Representations of Compact Lie Groups. Springer, New York.CrossRefzbMATHGoogle Scholar
  4. Cicogna, G. (1981). Symmetry breakdown from bifurcation. Lettre al Nuovo Cimento, 31, 600–602.MathSciNetCrossRefGoogle Scholar
  5. Duncan, K. and Eilbeck, J. C. (1987). Numerical studies of symmetry-breaking bifurcations in reaction-diffusion systems. In Proceedings of the International Workshop on Biomathematics and Related Computational Problems, ed. L.M. Ricciardi, Reidel, Dordrecht.Google Scholar
  6. Fraleigh, J. B. (1977). A First Course in Abstract Algebra, 2nd ed. Addison-Wesley, London.zbMATHGoogle Scholar
  7. Golubitsky, M., Stewart, I. and Schaeffer, D. G. (1988). Singularities and Groups in Bifurcation Theory, Vol. II. Appl. Math. Sci. 69, Springer, New York.CrossRefzbMATHGoogle Scholar
  8. Healey, T. J. (1988). Global bifurcation and continuation in the presence of symmetry with an application to solid mechanics. SIAM J. Math. Anal. 19, 824–840.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Lauterbach, R. (1986). An example of symmetry-breaking with submaximal isotropy. In Multiparameter Bifurcation Theory, eds. Golubitsky, M. and Guckenheimer, J., Contemp. Math. 56, 217–222, AMS, Providence.CrossRefGoogle Scholar
  10. Scovel, J. C., Kevrekidis, I. G. and Nicolaenko, B. (1988). Scaling laws and the prediction of bifurcation in systems modelling pattern formation. Phys. Letts. A 130, 73–80.CrossRefGoogle Scholar
  11. Vanderbauwhede, A. (1982). Local Bifurcation and Symmetry, Research Notes in Mathematics 75, Pitman, London.zbMATHGoogle Scholar
  12. Werner, B. and Spence, A. (1984). The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • P. J. Aston

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