Multiple homoclinics for a class of singular Hamiltonian systems. (English) Zbl 0887.58017

The authors investigate a second order Hamiltonian system of the form \(\ddot u + \nabla V(u) = 0\) in \(\mathbb{R}^n\), where the potential has a unique strict global maximum at the origin \(p\) and a singular set \(S \not\ni p\) such that \(\mathbb{R}^n \backslash S\) is open, path-connected and has non-trivial fundamental group \(\pi_1 = G\). Some additional conditions on behaviour of \(V\) near \(S\) and \(p\) are also assumed so that the class of systems under consideration includes, for instance, the classical \(n\)-body problem. Under some suitable conditions on \(V\) and \(G\) the authors prove the existence of multiple homoclinics, each one belonging to a different class of \(G\).


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70F10 \(n\)-body problems
Full Text: DOI


[1] Ambrosetti, A.; Zelati, V.Coti, Multiple homoclinic orbits for a class of conservative systems, Rend. sem. mat. univ. Padova, 89, 177-194, (1993) · Zbl 0806.58018
[2] Bertotti, M.L.; Bolotin, S.V., A variational approach for homoclinics in almost periodic Hamiltonian systems, Comm. appl. nonlinear anal., 2, 43-57, (1995) · Zbl 0858.34039
[3] M. L. Bertotti, L. Jeanjean, Multiplicity of homoclinic solutions for singular second order conservative systems, Proc. Roy. Soc. Edinburgh Sect. A · Zbl 0868.34001
[4] Bessi, U., Multiple homoclinic orbits for autonomous singular potentials, Proc. roy. soc. Edinburgh sect. A, 124, 785-802, (1994) · Zbl 0812.58088
[5] Bolotin, S.V., Existence of homoclinic motions, Vestnik moskov. univ. ser. I mat. mekh., 6, 98-103, (1980)
[6] Bolotin, S.V., Variational criteria for non integrability and chaos in Hamiltonian systems, Hamiltonian systems: integrability and chaotic behavior, (1994), Kluwer Academic Dordrecht/Norwell
[7] S. V. Bolotin, P. Negrini, Variational criterion for nonintegrability of a double pendulum · Zbl 0951.37029
[8] B. Buffoni, Infinitely many large amplitude homoclinic orbits for a class of autonomous Hamiltonian systems, J. Differential Equations · Zbl 0832.34032
[9] B. Buffoni, E. Séré, A global condition for quasi random behavior in a class of conservative systems · Zbl 0860.58027
[10] P. Caldiroli, L. Jeanjean, Homoclinics and heteroclinics for a class of conservative singular Hamiltonian systems, J. Differential Equation · Zbl 0887.34044
[11] P. Caldiroli, M. Nolasco, Multiple homoclinic solutions for a class of autonomous singular systems inR2, Ann. Inst. H. Poincare Anal. Non Linéaire · Zbl 0907.58014
[12] K. Cieliebak, E. Séré, Pseudo-holomorphic curves and multiplicity of homoclinic orbits · Zbl 0998.37500
[13] Zelati, V.Coti; keland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050
[14] Zelati, V.Coti; Rabinowitz, P.H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. amer. math. soc., 4, 693-727, (1991) · Zbl 0744.34045
[15] Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. amer. math. soc., 204, 113-135, (1975) · Zbl 0276.58005
[16] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983), Springer-Verlag Berlin/New York · Zbl 0515.34001
[17] Hedlund, G.A., The dynamics of geodesic flows, Bull. amer. math. soc., 45, 241-260, (1939) · JFM 65.0793.02
[18] Lions, P.L., The concentration – compactness principle in the calculus of variations, Rev. mat. iberoamericana, 1, 145-201, (1985) · Zbl 0704.49005
[19] Melnikov, V.K., On the stability of the center for periodic perturbations, Trans. Moscow math. soc., 12, 1-57, (1963)
[20] Meyer, K.R.; Sell, G.R., Melnikov transformations, Bernoulli bundles and almost periodic perturbations, Trans. amer. math. soc., 314, 63-105, (1989) · Zbl 0707.34041
[21] P. Montecchiari, M. Nolasco, S. Terracini, Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems · Zbl 0886.58014
[22] Morse, M., A fundamental class of geodesics in any closed surface of genus greater than one, Trans. amer. math. soc., 26, 25-61, (1924) · JFM 50.0466.04
[23] H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, 1897, 1899, Gauthier-Villars, Paris · JFM 30.0834.08
[24] P. H. Rabinowitz, Homoclinics for a singular Hamiltonian system, Geometric Analysis and the Calculus of Variations, J. Jost, International Press
[25] P. H. Rabinowitz, Homoclinics for an almost periodically forced Hamiltonian system, Topol. Methods Nonlinear Analysis
[26] Rabinowitz, P.H., Multibump solutions for an almost periodically forced singular Hamiltonian system, Electron. J. differential equations, 1995, (1995) · Zbl 0828.34034
[27] Rotman, J.J., An introduction to algebraic topology, Grad. texts in math., 119, (1993) · Zbl 0661.55001
[28] Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27-42, (1992) · Zbl 0725.58017
[29] Séré, E., Looking for the Bernoulli shift, Ann. inst. H. Poincaré anal. non linéaire, 10, 561-590, (1993) · Zbl 0803.58013
[30] 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
[31] Tanaka, K., A note on the existence of multiple homoclinic orbits for a perturbed radial potential, Nonlinear differential equations appl., 1, 149-162, (1994) · Zbl 0819.34032
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