Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits. (English) Zbl 0787.34041

Tanaka, Kazunaga; Mischaikow, Konstantin

J. Differ. Equations 94, No. 2, 315-339 (1991).

Homoclinic orbits for a time-periodic Hamiltonian system \((*)\) \(\dot z =JH_ z(t,z)\), \(H={1 \over 2}\langle Az,z \rangle+W(T,z)\) 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 \((*)\) 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.

Reviewer: J.Pöschl (Zürich)

MSC:

34C37

Homoclinic and heteroclinic solutions to ordinary differential equations

Keywords:

homoclinic orbits; periodic solutions; time-periodic Hamiltonian system

References:

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