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Zbl 0672.34037
Li, Shujie; Liu, J.Q.
Morse theory and asymptotic linear Hamiltonian system. (English)
[J] J. Differ. Equations 78, No.1, 53-73 (1989). ISSN 0022-0396

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 $-J\dot z=H'(z,t)$ $(*)\quad \vert H'(s,t)-h\sb{\infty}s\vert /\vert s\vert \to 0,$ as $\vert s\vert \to \infty$ $\vert H'(s,t)-h\sb 0s\vert /\vert s\vert \to 0,$ as $\vert s\vert \to 0$ $(h\sb{\infty},h\sb 0$ are constant symmetric matrices and $H'$ denotes the gradient of H with respect to the first 2n variables). Given a constant 2n$\times 2n$ matrix h we define an index $i\sp-(h)$. measuring the difference between the ``sizes'' of the negative spaces of the operators -Jd/dt-h and -Jd/dt. We define $i\sp 0(h)$ which is just the dimension of the null space of -Jd/dt-h. Let $H: R\sp{2n}\times R\to R$ be $C\sp 1,2\pi$-periodic in t and satisfying (*). If $i\sp 0(h\sb 0)=i\sp 0(h\sb{\infty})=0$ and $i\sp-(h\sb 0)\ne i\sp-(h\sb{\infty})$ then there exists at least one nontrivial periodic solution''. Morse theory and Galerkin method are combined.
[A.Halanay]

MSC 2000:: *34C25 Periodic solutions of ODE
70H05 Hamilton's equations

Keywords: Hamiltonian systems; Morse theory; Galerkin method

Cited in: Zbl 0926.34034 Zbl 0944.34034

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