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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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