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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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