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On the existence of homoclinic orbits on Riemannian manifolds

Published online by Cambridge University Press:  19 September 2008

Fabio Giannoni
Fabio Giannoni
Affiliation:
Center for the Mathematical Sciences, University of Wisconsin, Madison, WI53705, USA, Istituto di Matematiche Applicate ‘U.Dini’, University of Pisa, 56126 Pisa, Italy
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Abstract

We prove the existence of a non-trivial homoclinic orbit on a Riemannian manifold (possibly non-compact), for Hamiltonian systems of the second order of the form:

where the potential V is T-periodic in the time variable.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 1 , March 1994 , pp. 103 - 127
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

[1]Benci, V. & Giannoni, F.. Homoclinic orbits on compact manifolds. J. Math. Anal. Appl. 157 (1991), 568–576.CrossRefGoogle Scholar
[2]Canino, A.. On p-convex sets and geodesics. J. Diff. Eq. 75 (1988), 118–157.CrossRefGoogle Scholar
[3]Coti-Zelati, V., Ekeland, I. & Séré, E.. A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288 (1990), 133–160.CrossRefGoogle Scholar
[4]Coti-Zelati, V. & Rabinowitz, P. H.. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991), 693–727.CrossRefGoogle Scholar
[5]Fadell, E. & Husseini, S.. Category of loop space of open subsets in Euclidean space. Nonlin. Anal. T.M.A. 17 (1991), 1163–1175.CrossRefGoogle Scholar
[6]Giannoni, F.. Multiplicity of principal bounce trajectories on Riemannian manifolds. Diff. and Int. Eq. 6 (1993), 1451–1480.Google Scholar
[7]Giannoni, F. & Rabinowitz, P. H.. On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of second order Hamiltonian systems. Nonl. Diff. Eq. Appl. 1 (1994), 1–46.CrossRefGoogle Scholar
[8]Hofer, H. & Wysocki, K.. First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann. 288 (1990), 133–160.CrossRefGoogle Scholar
[9]Milnor, J.. Morse theory. Ann. Math. Studies 51 Princeton University Press: Princeton, 1963.Google Scholar
[10]Nash, J.. The embedding problem for Riemannian manifolds. Ann. of Math. 63 (1956), 20–63.CrossRefGoogle Scholar
[11]Palais, R. S.. Morse theory on Hilbert manifolds. Topology 2 (1963), 299–340.CrossRefGoogle Scholar
[12]Rabinowitz, P. H.. Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. 65 Amer. Math. Soc.: Providence, 1986.CrossRefGoogle Scholar
[13]Rabinowitz, P. H.. Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinburgh 114A (1990), 33–38.Google Scholar
[14]Rabinowitz, P. H. & Tanaka, K.. Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206 (1991), 473–199.CrossRefGoogle Scholar
[15]Schwartz, J. T.. Nonlinear Functional Analysis. Gordon and Breach: New York, 1969.Google Scholar
[16]Séré, E.. Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209 (1992), 27–2.CrossRefGoogle Scholar
[17]Tanaka, K.. Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits. J. Diff. Eq. 94 (1991), 315–339.CrossRefGoogle Scholar