Spectral flow and bifurcation of critical points of strongly indefinite functionals part II. Bifurcation of periodic orbits of hamiltonian systems

摘要

Our main results here are as follows: Let X-perpendicular to he a family of 2 pi-periodic Hamiltonian vectorfields that depend smoothly on a real parameter lambda in [a, b] and has a known. trivial. branch s(lambda) of 2 pi-periodic solutions. Let P-lambda be the Pioncare map of the linearization of X-lambda at s(lambda). If the Conley-Zehnder index of the path P-lambda does not vanish, then any neighborhood of the trivial branch of periodic solutions contains 2 pi-periodic solulions not on the branch. Moreover, if each solution s(lambda) is constant and each linearization A(lambda) of X-lambda at s(lambda) is time independent then bifurcation of 2 pi-periodic orbits from the branch of equilibria arises whenever i(A(b)) not equal i(A(b)), where i(A) is the index of the linear Hamiltonian system Ju = Au. (C) 2000 Academic Press. [References: 22]