In this work we prove the lower bound for the number of T-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, T-periodic in time, with T-Maslov indices i_0, i_∞ at the origin and at infinity, has at least |i_∞ - i_0| periodic solutions, and an additional one if i_0 is even. Our argument combines the Poincaré-Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.
Lower bound on the number of periodic solutions for asymptotically linear planar hamiltonian systems
Gidoni Paolo
;
2019-01-01
Abstract
In this work we prove the lower bound for the number of T-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, T-periodic in time, with T-Maslov indices i_0, i_∞ at the origin and at infinity, has at least |i_∞ - i_0| periodic solutions, and an additional one if i_0 is even. Our argument combines the Poincaré-Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.File in questo prodotto:
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