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Morse theory and asymptotic linear Hamiltonian system. (English) Zbl 0672.34037
We try to quote from the Introduction in order to give an idea on the paper avoiding technical details. “We consider the existence of $2\pi$- periodic solutions of Hamiltonian systems $-J\stackrel{˙}{z}={H}^{\text{'}}\left(z,t\right)$ $\left(*\right)\phantom{\rule{1.em}{0ex}}|{H}^{\text{'}}\left(s,t\right)-{h}_{\infty }s|/|s|\to 0,$ as $|s|\to \infty$ $|{H}^{\text{'}}\left(s,t\right)-{h}_{0}s|/|s|\to 0,$ as $|s|\to 0$ $\left({h}_{\infty },{h}_{0}$ are constant symmetric matrices and ${H}^{\text{'}}$ denotes the gradient of H with respect to the first 2n variables). Given a constant 2n$×2n$ matrix h we define an index ${i}^{-}\left(h\right)$. measuring the difference between the “sizes” of the negative spaces of the operators -Jd/dt-h and -Jd/dt. We define ${i}^{0}\left(h\right)$ which is just the dimension of the null space of -Jd/dt-h. Let $H:{R}^{2n}×R\to R$ be ${C}^{1},2\pi$-periodic in t and satisfying (*). If ${i}^{0}\left({h}_{0}\right)={i}^{0}\left({h}_{\infty }\right)=0$ and ${i}^{-}\left({h}_{0}\right)\ne {i}^{-}\left({h}_{\infty }\right)$ then there exists at least one nontrivial periodic solution”. Morse theory and Galerkin method are combined.
Reviewer: A.Halanay

##### MSC:
 34C25 Periodic solutions of ODE 70H05 Hamilton’s equations
##### Keywords:
Hamiltonian systems; Morse theory; Galerkin method