Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

  • Jacobo PejsachowiczEmail author
  • Nils Waterstraat


Given a continuous family of C 2 functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.

Mathematics Subject Classification

Primary 58E07 Secondary 47J15 37J45 37J20 


Bifurcation Fredholm functional spectral flow Hamiltonian systems periodic solutions 


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  1. APS76.
    Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc. 79, 71–99 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Ba93.
    T. Bartsch, Topological Methods for Variational Problems with Symmetries. Lecture Notes in Math. 1560, Springer, Berlin, 1993Google Scholar
  3. Ba92.
    Bartsch T.: The Conley index over a space. Math. Z. 209, 167–177 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. BW85.
    Booss B., Wojciechowski K.: Desuspension of splitting elliptic symbols. I. Ann. Global Anal. Geom. 3, 337–383 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  5. CFP00.
    Ciriza E., Fitzpatrick P.M., Pejsachowicz J.: Uniqueness of spectral flow. Math. Comput. Modelling 32, 1495–1501 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. C78.
    C. Conley, Isolated Invariant Sets and the Morse Index. CBMS Reg. Conf. Ser. Math. 38, Amer. Math. Soc., Providence, RI, 1978.Google Scholar
  7. CL88.
    S.-N. Chow and R. Lauterbach. A bifurcation theorem for critical points of variational problems. Nonlinear Anal. 12 (1988), 51–61.Google Scholar
  8. CLM94.
    Cappell S.E., Lee R., Miller E.: On the Maslov index. Comm. Pure Appl. Math. 47, 121–186 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Fl88.
    Floer A.: An instanton-invariant for 3-manifolds. Comm. Math. Phys. 118, 215–240 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. FPS.
    P. M. Fitzpatrick, J. Pejsachowicz and C. A. Stuart, Spectral flow for paths of unbounded operators and bifurcation of critical points. Preprint.Google Scholar
  11. FPR99.
    P. M. Fitzpatrick, J. Pejsachowicz and L. Recht, Spectral flow and bifurcation of critical points of strongly indefinite functionals. I. General theory. J. Funct. Anal. 162 (1999), 52–95.MathSciNetCrossRefzbMATHGoogle Scholar
  12. FPR00.
    P. M. Fitzpatrick, J. Pejsachowicz and L. Recht, Spectral flow and bifurcation of critical points of strongly indefinite functionals. II. Bifurcation of periodic orbits of Hamiltonian systems. J. Differential Equations 163 (2000), 18–40.Google Scholar
  13. GG74.
    M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Grad. Texts in Math. 14, Springer, Berlin, 1973.Google Scholar
  14. HW48.
    W. Hurewicz and H. Wallmann, Dimension Theory, Princeton Math. Ser. 4, Princeton University Press, Princeton, NJ, 1941.Google Scholar
  15. Ka76.
    T. Kato, Perturbation theory for linear operators. 2nd ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1976.Google Scholar
  16. K04.
    H. Kielhöfer, Bifurcation Theory–An Introduction with Applications to PDEs. Springer, New York, 2004.Google Scholar
  17. Kr64.
    M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford, 1964Google Scholar
  18. MPP05.
    M. Musso, J. Pejsachowicz and A. Portaluri, A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds. Topol. Methods Nonlinear Anal. 25 (2005), 69–99.Google Scholar
  19. MPP07.
    M. Musso, J. Pejsachowicz and A. Portaluri, Morse index and bifurcation of p-geodesics on semi Riemannian manifolds. ESAIM Control Optim. Calc. Var. 13 (2007), 598–621Google Scholar
  20. Ni93.
    Nicolaescu L.: The Maslov index, the spectral flow and splittings of manifolds. C. R. Acad. Sci. Paris Sér. I Math. 317, 515–519 (1993)MathSciNetzbMATHGoogle Scholar
  21. Ph96.
    Phillips J.: Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull. 39, 460–467 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. PW13.
    A. Portaluri and N. Waterstraat, Bifurcation results for critical points of families of functionals. Differential Integral Equations, to appear.Google Scholar
  23. RS95.
    J. Robbin and D. Salamon, The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27 (1995), 1–33.Google Scholar
  24. W08.
    C. Wahl, A new topology on the space of unbounded self-adjoint operators, K-theory and spectral flow. In: C*-Algebras and Elliptic Theory II, Trends Math., Birkhäuser, Basel, 2008, 297–309.Google Scholar
  25. Wa13.
    N. Waterstraat, A remark on the space of metrics having nontrivial harmonic spinors. J. Fixed Point Theory Appl. 13 (2013), 143–149.MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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