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 \((*)\) \(\dot z =JH_ z(t,z)\), \(H={1 \over 2}\langle Az,z \rangle+W(T,z)\) 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.


34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
Full Text: DOI


[1] Benci, V.; Giannoni, F., Homoclinic orbits on compact manifolds, J. Math. Anal. Appl., 157, 568-576 (1991) · Zbl 0737.58052
[2] Coti-Zalati, V.; Ekeland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288, 133-160 (1990) · Zbl 0731.34050
[3] Edwards, R. E.; Gaudry, G. I., Littlewood-Paley and multiplier theory, (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 90 (1977), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · Zbl 0464.42013
[4] Felmer, P. L., Subharmonic solutions near an equilibrium point for Hamiltonian systems, Manuscripta Math., 66, 359-396 (1990) · Zbl 0688.34027
[5] Felmer, P. L., Heteroclinic orbits for spatially periodic Hamiltonian systems (1989), preprint · Zbl 0749.58021
[6] Hofer, H.; Wysocki, K., First order elliptic system and the existence of homoclinic orbits in Hamiltonian system, Math. Ann., 288, 483-503 (1990) · Zbl 0702.34039
[7] Rabinowitz, P. H., On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 33, 609-633 (1980) · Zbl 0425.34024
[8] Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, (CBMS Regional Conference Series in Mathematics, Vol. 65 (1986), Amer. Math. Soc: Amer. Math. Soc Providence), 21
[9] Rabinowitz, P. H., Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6, 331-346 (1989) · Zbl 0701.58023
[10] Rabinowitz, P. H., Homoclinic orbits for a class of Hamiltonian systems, (Proc. Roy. Soc. Edinburgh, 114A (1990)), 33-38 · Zbl 0705.34054
[11] Rabinowitz, P. H.; Tanaka, K., Some results on connecting orbits for a class of Hamiltonian system, Math. Zeit., 206, 473-499 (1991) · Zbl 0707.58022
[12] Tanaka, K., Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7, 427-438 (1990) · Zbl 0712.58026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.