# American Institute of Mathematical Sciences

January  2019, 39(1): 585-606. doi: 10.3934/dcds.2019024

## Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems

 1 Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy 2 Centro de Matemática, Aplicações Fundamentais e Investigação Operacional (CMAF – CIO), Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edificio C6, 1749–016 Lisboa, Portugal 3 Departamento de Matemática and Centro de Matemática, Aplicações Fundamentais, e Investigação Operacional (CMAF – CIO), Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edificio C6, 1749–016 Lisboa, Portugal

* Corresponding author

Received  May 2018 Published  October 2018

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.

Citation: Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024
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##### References:
An Hamiltonian flow corresponding to case ⅰ) of Remark 14. For small periods $T$, the system has $i_0=0$, $i_\infty=-1$, and the only non-zero $T$-periodic orbits are the fixed points $A$ and $B$
An Hamiltonian flow corresponding to case ⅱ) of Remark 14. For small periods $T$, the system has $i_0=1$, $i_\infty=-1$, and the only non-zero $T$-periodic orbits are the fixed points $M$ and $S$
An Hamiltonian flow corresponding to case ⅲ) of Remark 14. For small periods $T$, the system has $i_0=1$, $i_\infty=0$, and the only non-zero $T$-periodic orbit is the fixed point $S$
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