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Homoclinic orbits for a class of Hamiltonian systems

Published online by Cambridge University Press:  14 November 2011

Paul H. Rabinowitz
Paul H. Rabinowitz
Affiliation:
Mathematics Department and Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, Wisconsin 53705, U.S.A.
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Synopsis

Consider the second order Hamiltonian system:

where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 1-2 , 1990 , pp. 33 - 38
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Coti-Zelati, V. and Ekeland, I.. A variational approach to homoclinic orbits in Hamiltonian systems (preprint).Google Scholar
2Rabinowitz, P. H.. Periodic and heteroclinic orbits for a periodic Hamiltonian system. Anal. Nonlineaire 6 (1989), 331–346.Google Scholar
3Benci, V. and Giannoni, F.. Homoclinic orbits on compact manifolds (preprint).Google Scholar
4Tanaka, K.. Homoclinic orbits for a singular second order Hamiltonian system (preprint).Google Scholar
5Hofer, H. and Wysocki, K.. First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems (preprint).Google Scholar
6Lions, P. L.. The concentration-compactness principle in the calculus of variations. Rev. Mat. Iberoamericana 1 (1985), 145–201.CrossRefGoogle Scholar
7Floer, A., Hofer, H. and Viterbo, C.. The Weinstein conjecture in P × C 1 (preprint).Google Scholar
8Rabinowitz, P. H.. Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31 (1978), 36–68.CrossRefGoogle Scholar
9Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Fund. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar