Topological Invariants of Bifurcation

  • Jacobo Pejsachowicz
Part of the Trends in Mathematics book series (TM)


I will shortly discuss an approach to bifurcation theory based on elliptic topology. The main goal is a construction of an index of bifurcation points for C 1-families of Fredholm maps derived from the index bundle of the family of linearizations along the trivial branch. As illustration, I will present an application to bifurcation of homoclinic solutions of non-autonomous differential equations from a branch of stationary solutions.


Bifurcation Fredholm maps Index bundle J-homomorphism 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.F. Adams, On the groups J(X)-I. Topology 2 (1963), 181–195.CrossRefMathSciNetGoogle Scholar
  2. [2]
    J.C. Alexander, Bifurcation of zeroes of parametrized functions. J. of Funct. Anal, 29 (1978), 37–53.zbMATHCrossRefGoogle Scholar
  3. [3]
    J.C. Alexander, James Yorke, Calculating bifurcation invariants as elements in the homotopy of the general linear group. J. of Pure and Appl. Algebra, 13 (1978), 1–9.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    M.F. Atiyah, K-Theory. Benjamin, 1967.Google Scholar
  5. [5]
    M.F. Atiyah, Thom complexes. Proc. Lond. Math. Soc. 11 (1961), 291–310.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    T. Bartsch, The global structure of the zero set of a family of semilinear Fredholm maps. Nonlinear Analysis 17 (1991), 313–331.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    T. Bartsch, A global index for bifurcation of fixed points. J. reine angew. Math. 391 (1988), 181–197.zbMATHMathSciNetGoogle Scholar
  8. [8]
    P.M. Fitzpatrick, J. Pejsachowicz, Fundamental group of the space of Fredholm operators and global analysis of non linear equations. Contemporary Math. 72 (1988), 47–87.MathSciNetGoogle Scholar
  9. [9]
    P.M. Fitzpatrick, J. Pejsachowicz, Nonorientability of the index bundle and severalparameter bifurcation. J. of Functional Anal. 98 (1991), 42–58.zbMATHCrossRefGoogle Scholar
  10. [10]
    P.M. Fitzpatrick, I. Massabò, J. Pejsachowicz, Global several parameter bifurcation and continuation theorems. Math. Ann. 263 (1983), 61–73.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    P.M. Fitzpatrick, J. Pejsachowicz, P.J. Rabier, The degree for proper C 2-Fredholm mappings I. J. reine angew. Math. 424 (1992), 1–33.MathSciNetGoogle Scholar
  12. [12]
    N. Hitchin,Harmonic spinors, Adv. in Math. 14 (1974), 1–55.Google Scholar
  13. [13]
    J. Ize, Bifurcation theory for Fredholm operators. Mem. Am. Math.Soc. 174 (1976).Google Scholar
  14. [14]
    J. Ize, Necessary and sufficient conditions for multiparameter bifurcation. Rocky Mountain J. of Math 18 (1988), 305–337.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Ize, Topological bifurcation. Topological Nonlinear Analysis, Birkhäuser, Progress in nonlinear differential equations, 15 (1995), 341–463.MathSciNetGoogle Scholar
  16. [16]
    K. Jänich, Vektorraumbündel und der Raum der Fredholm-Operatoren. Mathematische Annalen, 161 (1965),129–142.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, 1976.Google Scholar
  18. [18]
    H. Kielhöfer, Multiple eigenvalue bifurcation for Fredholm mappings. J. reine angew. Math. 358 (1985), 104–124.zbMATHMathSciNetGoogle Scholar
  19. [19]
    J. Pejsachowicz, K-theoretic methods in bifurcation theory. Contemporary Math., 72 (1988), 193–205.MathSciNetGoogle Scholar
  20. [20]
    J. Pejsachowicz, The Leray-Schauder Reduction and Bifurcation for Parametrized Families of Nonlinear Elliptic Boundary Value Problems. TMNA 18 (2001), 243–268.zbMATHMathSciNetGoogle Scholar
  21. [21]
    J. Pejsachowicz, The index bundle and bifurcation theory of Fredholm maps. In preparation.Google Scholar
  22. [22]
    J. Pejsachowicz, P.J. Rabier, Degree theory for C 1-Fredholm mappings of index 0. Journal d’Analyse Mathématique 76 (1998), 289–319.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    C. Vafa, E. Witten, Eigenvalue inequalities for fermions in gauge theories. Comm. Math. Phys. 95 (1984), 257–276.CrossRefMathSciNetGoogle Scholar
  24. [24]
    V.G. Zvyagin, On oriented degree of a certain class of perturbations of Fredholm mappings and on bifurcations of solutions of a nonlinear boundary value problem with noncompact perturbations. Mat. USSR Sbornik 74 (1993), 487–512.CrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Jacobo Pejsachowicz
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorino, ToItaly

Personalised recommendations