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Subharmonic solutions of indefinite Hamiltonian systems via rotation numbers. (English) Zbl 1479.37066

Summary: We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems \(Jz^{\prime}=\nabla H(t,z)\) from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on \(\frac{\partial H}{\partial x}(t,x,y)\), uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré-Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

MSC:

37J46 Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems
37J51 Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods
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