Skip to main content
SpringerLink
Log in

Group-Theoretic Bifurcation Theory

  • Chapter
  • First Online:
Imperfect Bifurcation in Structures and Materials

Part of the book series: Applied Mathematical Sciences ((AMS,volume 149))

  • 590 Accesses

Abstract

Group-theoretic bifurcation theory is introduced as a means to describe qualitative aspects of symmetry-breaking bifurcation, such as possible types of critical points and the symmetry of bifurcating solutions. We advance a series of mathematical concepts and tools, including: group equivariance, Liapunov–Schmidt reduction, equivariant branching lemma, and block-diagonalization. The theory of linear representations of finite groups in Chap. 7 forms a foundation of this chapter. This chapter is an extension of Chap. 2 to a system with symmetry and a prerequisite to the study of structures and materials with dihedral symmetry in Chaps. 9–13 and larger symmetries in Chaps. 14–17.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 84.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See, for example, Sattinger, 1979 [167], 1980 [168]; Golubitsky and Schaeffer, 1985 [55]; and Golubitsky, Stewart, and Schaeffer, 1988 [57].

  2. 2.

    See, for example, Olver, 1986 [151], 1995 [152]; Mitropolsky and Lopatin, 1988 [133]; Allgower, Böhmer, and Golubitsky, 1992 [1]; Marsden and Ratiu, 1994 [128]; and Hoyle, 2006 [69].

  3. 3.

    Loosely speaking, the term “generically” might be replaced by “unless the parameters take special values.”

  4. 4.

    Reciprocity plays a significant role in the bifurcation analysis of Cn-symmetric systems (cf., Sect. 9.9).

  5. 5.

    Since P is the projection on an M-dimensional subspace, Eq. (8.36) represents M constraints and (8.37) represents (N − M) constraints.

  6. 6.

    Since T is assumed to be unitary, U and V  in (8.45) are valid choices (cf., Remark 8.4).

  7. 7.

    We can replace T(g)w by w since w is arbitrary in \(\mathrm {ker} (J^{0}_{\mathrm {c}} )\) and \(\mathrm {ker} (J^{0}_{\mathrm {c}} )\) is G-invariant. We can also replace S(g)v by v since v is arbitrary.

  8. 8.

    The uniqueness assertion applies since \( T(g) \boldsymbol {\varphi } ( T(g^{-1})\boldsymbol {w}, \tilde {f}, S(g^{-1})\boldsymbol {v}) \in U\) by the G-invariance of U and S(g −1)v stays in a neighborhood of v 0 by ∥S(g −1)v −v 0∥ = ∥S(g −1)(v −v 0)∥ = ∥v −v 0∥. Here the first equality holds by (8.32) and the second equality by the unitarity of S.

  9. 9.

    For a block matrix \(\overline {J}\) as in (8.56), the matrix \(\overline {J}_{[1,1]} - \overline {J}_{[1,2]} (\overline {J}_{[2,2]})^{-1} \overline {J}_{[2,1]} \) is called the Schur complement .

  10. 10.

    This is true for a certain type of double bifurcation point of a Dn-symmetric system (cf., (9.60) with \(\hat {n}\ge 4\)).

  11. 11.

    Equation (8.82) corresponds to the bifurcation equation (8.26) for the perfect system with v = v 0, where the imperfection parameter vector v is suppressed.

  12. 12.

    See Lemma 7.1(iii) (Schur’s lemma) in Sect. 7.3.4.

  13. 13.

    This counterclockwise rotation appears to be clockwise in Fig. 8.2 since the z-axis is directed downward.

  14. 14.

    In Sect. 9.2.2 we introduce a more systematic notation: \(\mu _{1} = (+,+)_{\mathrm {D}_{3}}\), \(\mu _{2} = (+,-)_{\mathrm {D}_{3}}\), and \(\mu _{3} = (1)_{\mathrm {D}_{3}}\). Every μ i is absolutely irreducible, therefore, R a(D3) = R(D3).

References

  1. Allgower, E., Böhmer, K., Golubitsky M. eds. (1992) Bifurcation and Symmetry. International Series of Numerical Mathematics, Vol. 104. Birkhäuser, Basel.

    Google Scholar 

  2. Cicogna, G. (1981) Symmetry breakdown from bifurcations. Lettere al Nuovo Cimento 31, 600–602.

    Article  MathSciNet  Google Scholar 

  3. Fujii, H., Mimura, M., Nishiura, Y. (1982) A picture of the global bifurcation diagram in ecological interacting and diffusing systems. Physica D 5 (1), 1–42.

    Article  MathSciNet  Google Scholar 

  4. Golubitsky, M., Schaeffer, D.G. (1985) Singularities and Groups in Bifurcation Theory, Vol. 1. Applied Mathematical Sciences, Vol. 51. Springer-Verlag, New York.

    Book  Google Scholar 

  5. Golubitsky, M., Stewart, I., Schaeffer, D.G. (1988) Singularities and Groups in Bifurcation Theory, Vol. 2. Applied Mathematical Sciences, Vol. 69. Springer-Verlag, New York.

    Book  Google Scholar 

  6. Hoyle, R. (2006) Pattern Formation: An Introduction to Methods. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK.

    Book  Google Scholar 

  7. Keener, J.P. (1979) Secondary bifurcation and multiple eigenvalues. SIAM J. Appl. Math. 37 (2), 330–349.

    Article  MathSciNet  Google Scholar 

  8. Marsden, J.E., Ratiu, T.S. (1994) Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, Vol. 17. Springer-Verlag, New York; 2nd ed., 1999.

    Google Scholar 

  9. Mitropolsky, Yu. A., Lopatin A. K. (1988) Nonlinear Mechanics, Groups and Symmetry. Mathematics and Its Applications. Kluwer Academic, Dordrecht.

    Google Scholar 

  10. Olver, P.J. (1986) Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, Vol. 107. Springer-Verlag, New York; 2nd ed., 2000.

    Google Scholar 

  11. Olver, P.J. (1995) Equivariance, Invariants, and Symmetry. Cambridge University Press, Cambridge, UK.

    Book  Google Scholar 

  12. Sattinger, D.H. (1979) Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Mathematics, Vol. 762. Springer-Verlag, Berlin.

    Google Scholar 

  13. Sattinger, D.H. (1980) Bifurcation and symmetry breaking in applied mathematics. Bull. Amer. Math. Soc. 3 (2), 779–819.

    Article  MathSciNet  Google Scholar 

  14. Thompson, J.M.T., Hunt, G.W. (1984) Elastic Instability Phenomena. Wiley, Chichester.

    MATH  Google Scholar 

  15. Vanderbauwhede, A. (1980) Local Bifurcation and Symmetry. Habilitation Thesis, Rijksuniversiteit Gent.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Tohoku University, Sendai, Japan

    Kiyohiro Ikeda

  2. Department of Economics and Business Administration, Tokyo Metropolitan University, Hachioji, Japan

    Kazuo Murota

Authors
  1. Kiyohiro Ikeda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kazuo Murota
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ikeda, K., Murota, K. (2019). Group-Theoretic Bifurcation Theory. In: Imperfect Bifurcation in Structures and Materials. Applied Mathematical Sciences, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-030-21473-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation