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

  • Jacobo PejsachowiczEmail author
  • Nils Waterstraat
Article

Abstract

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 

Keywords

Bifurcation Fredholm functional spectral flow Hamiltonian systems periodic solutions 

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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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