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Periodic solutions for a class of nonautonomous Hamiltonian systems. (English) Zbl 0960.37028

The authors deal with the existence of periodic solutions for a Hamiltonian system \[ J\dot z- B(t)z= \nabla H(t,z), \quad z\in \mathbb{R}^{2N}, \quad t\in \mathbb{R}, \tag{1} \] where \(B(t)\) is a given \(T\)-periodic and symmetric \(2N\times 2N\) matrix function of class \(C^1\) in \(t\), \(H\in C^1 (\mathbb{R}\times \mathbb{R}^{2N}, \mathbb{R})\) is \(T\)-periodic in \(t\), \(\nabla H:= \nabla _z H\in C(\mathbb{R}\times \mathbb{R}^{2N}, \mathbb{R}^{2N})\) and \(J\) is the standard symplectic matrix. Under some natural conditions on \(H\), the authors prove the existence of \(T\)-periodic solutions of (1).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C25 Periodic solutions to ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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