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

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  • 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.

    Mathematics Subject Classification: 37J45, 34C25, 70H12.


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  • Figure 1.  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 $B$

    Figure 2.  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 $S$

    Figure 3.  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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