## Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity.(English)Zbl 0831.34046

This paper studies the asymptotically linear Hamiltonian systems (1) $$\dot z \equiv J H_z' (z,t)$$, $$z \in \mathbb{R}^{2n}$$, with commonly used notations. The function $$H$$ satisfies the conditions 1. $$H \in C^2 (\mathbb{R}^{2n} \times \mathbb{R}, \mathbb{R})$$ and $$H$$ is 1-periodic in $$t$$; 2. $$H_z' (z,t) = B_0 (t)z + o( |z |)$$ as $$|z |\to 0$$ uniformly in $$t$$; 3. $$H_z' (z,t) = B_\infty (t)z + o(|z |)$$ as $$|z |\to \infty$$ uniformly in $$t$$, where $$B_0 (t)$$ and $$B_\infty (t)$$ are symmetric matrices in $$\mathbb{R}^{2n}$$, and continuous and 1-periodic in $$t$$. Using the Maslov-index theory, the author proves the existence of 1-periodic solutions of (1) in the resonant case that $$B_\infty (t)$$ is finitely degenerate and time dependent. The result of this paper generalizes a recent work of K. C. Chang [Research Report, No. 30, Inst. of Math. and Dept. of Math., Peking University (1991)].

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37G99 Local and nonlocal bifurcation theory for dynamical systems
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