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Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits. (English) Zbl 0787.34041
Homoclinic orbits for a time-periodic Hamiltonian system $\left(*\right)$ $\stackrel{˙}{z}=J{H}_{z}\left(t,z\right)$, $H=\frac{1}{2}〈Az,z〉+W\left(T,z\right)$ are found, assuming that $z=0$ is a hyperbolic equilibrium point and that $W$ has global superquadratic growth in $z$. They are obtained as local ${C}^{1}$- limits of certain nontrivial $T$-periodic solutions of $\left(*\right)$ as $T\to \infty$, where the hyperbolicity prevents them from shrinking to zero. This approach extends results by Rabinowitz for second order Hamiltonian systems, and it differs from corresponding results by Coti-Zelati, Ekeland & Séré and Hofer & Wysocki. The references are given in the paper.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE