Periodic solutions of second order Hamiltonian systems bifurcating from infinity

The goal of this article is to study closed connected sets of periodic solutions, of autonomous second order Hamiltonian systems, emanating from infinity. The main idea is to apply the degree for SO(2)-equivariant gradient operators defined by the second author in [S. Rybicki, SO(2)-degree for orthogonal maps and its applications to bifurcation theory, Nonlinear Anal. TMA 23 (1) (1994) 83–102]. Using the results due to Rabier [P. Rabier, Symmetries, topological degree and a theorem of Z.Q. Wang, Rocky Mountain J. Math. 24 (3) (1994) 1087–1115] we show that we cannot apply the Leray–Schauder degree to prove the main results of this article. It is worth pointing out that since we study connected sets of solutions, we also cannot use the Conley index technique and the Morse theory.

RésuméLe but de cet article est l'étude des ensembles fermés et connexes de solutions périodiques, émanant de l'infini, des systèmes hamiltoniens autonomes de second ordre. L'idée principale consiste à appliquer le degré aux opérateurs de gradient SO(2)-équivariants définis par le second auteur dans [S. Rybicki, SO(2)-degree for orthogonal maps and its applications to bifurcation theory, Nonlinear Anal. TMA 23 (1) (1994) 83–102]. Moyennant un résultat de Rabier [P. Rabier, Symmetries, topological degree and a theorem of Z.Q. Wang, Rocky Mountain J. Math. 24 (3) (1994) 1087–1115], on démontre que l'on ne peut pas appliquer le degré de Leray–Schauder pour obtenir le résultat principal de ce travail. Il est important de souligner que, vu que l'on étudie des ensembles connexes de solutions, ni la technique de l'indice de Conley, ni la théorie de Morse ne peuvent être appliquées ici.