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A “Birkhoff-Lewis” type result for a class of Hamiltonian systems. (English) Zbl 0616.58014
Consider the Hamiltonian system of differetial equations \[ (1)\quad - J\dot z=H_ z(t,z), \] where \(z=(p,q)\in {\mathbb{R}}^{2N}\), J denotes the symplectic structure in \({\mathbb{R}}^{2N}\) and \(H_ z\) denotes the gradient of the Hamiltonian function H with respect to z. Suppose that H is \(\tau\)-periodic in the t variable and \(H_ z(t,0)=0\). If H satisfies suitable assumptions it can be proved that (1) possesses a sequence \(z_ k\) of \(k\cdot \tau\) periodic solutions (k-prime) converging to zero uniformly in \(C^ 1({\mathbb{R}},{\mathbb{R}}^{2N})\).

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
58E30 Variational principles in infinite-dimensional spaces
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