Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals

By P. Fitzpatrick, J. Pejsachowicz, L. Recht. 2000

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Abstract Our main results here are as follows: Let X λ be a family of 2 π -periodic Hamiltonian vectorfields that depend smoothly on a real parameter λ in [ a ,  b ] and has a known, trivial, branch s λ of 2 π -periodic solutions. Let P λ be the Poincare map of the linearization of X λ at s λ . If the Conley–Zehnder index of the path P λ does not vanish, then any neighborhood of the trivial branch of periodic solutions contains 2 π -periodic solutions not on the branch. Moreover, if each solution s λ is constant and each linearization A λ of X λ at s λ is time independent, then bifurcation of 2 π -periodic orbits from the branch of equilibria arises whenever i ( A b )≠ i ( A b ), where i ( A ) is the index of the linear Hamiltonian system Ju = Au .

Subfields: Control and Systems Engineering, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1227cb28b053f061fbcd40748a7a1321c86a00e3