Morse theory for asymptotically linear Hamiltonian systems
The paper is concerned with Hamiltonian systems (HS) \(\dot{z}=JH_z(z,t)\) where \(H:\mathbb R^{2N}\times\mathbb R\rightarrow\mathbb R\) is of class \({\mathcal C}^2\) and \(T\)-periodic in \(t\). The author develops a Morse-Conley theory for the action integral \(f:E=H^{1/2}(\mathbb R/T\mathbb Z,\mathbb R^{2N})\rightarrow\mathbb R\) associated to (HS). Since \(f\) is strongly indefinite the usual Morse index of a critical point \(z\in E\) is replaced by a relative version which is equal to the Maslov index of \(z\) as a \(T\)-periodic solution of (HS). If (HS) is asymptotically linear then the theory implies a result of \textit{C. Conley} and \textit{E. Zehnder} [Commun. Pure Appl. Math. 37, 207-253 (1984; Zbl 0559.58019)]. The proof given here avoids the finite-dimensional reduction used by Conley and Zehnder. Another application of the theory assumes the existence of a \(T\)-periodic solution whose Maslov index does not lie in the range prescribed by the linearized system at infinity which is allowed to be resonant. In that case a second solution must exist.
scientific article; zbMATH DE number 1419635
scientific article