## Homoclinics and heteroclinics for a class of conservative singular Hamiltonian systems.(English)Zbl 0887.34044

The authors investigate by variational methods an autonomous Hamiltonian system where the potential has a strict global maximum and a singularity at some point.
They prove the existence of homoclinics which turn $$k$$ times around the singularity and infinitely many geometrically distinct homoclinics. Moreover, the existence of a heteroclinic orbit and periodic orbit, and the chaotic behavior on some invariant subsets of the energy level $$\varepsilon$$ (small) are given. The techniques used are quite manifold, and useful for the study of singular Hamiltonian systems.
Reviewer: Z.Jing (Beijing)

### MSC:

 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text:

### References:

 [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., 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., Variational criteria for nonintegrablity and chaos in Hamiltonian systems, Hamiltonian systems: integrability and chaotic behavior, NATO ASI series, Vol. 331, (1994) [6] S. v. Bolotin, P. Negrini, Variational criterion for nonintegrablity of a double pendulum · Zbl 0951.37029 [7] Bhatia, N.P.; Szego, G.P., Dynamical systems: stability theory and applications, Lecture notes in mathematics, Vol. 35, (1967), Springer-Verlag Berlin/New York [8] Buffoni, B.; Séré, E., A global condition for quasi random behavior in a class of conservative systems, Com. pure appl. math., 49, 285-305, (1996) · Zbl 0860.58027 [9] Caldiroli, P.; Montecchiari, P., Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. appl. nonlinear anal., 1, 97-129, (1994) · Zbl 0867.70012 [10] P. Caldiroli, M. Nolasco, Multiple homoclinic solutions for a class of autonomous singular systems inR2, Ann. Inst. H. Poincaré, Anal. Non Linéaire · Zbl 0907.58014 [11] Cieliebak, K.; Séré, E., Pseudo-holomorphic curves and multiplicity of homoclinic orbits, Duke math. J., 77, 483-518, (1995) · Zbl 0842.58022 [12] Coti Zelati, v.; Ekeland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050 [13] Coti Zelati, v.; Rabinowitz, P.H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. amer. math. soc., 4, 693-727, (1991) · Zbl 0744.34045 [14] Felmer, P.L., Heteroclinic orbits for spatially periodic Hamiltonian systems, Ann. inst. H. Poincaré, anal. non linéaire, 8, 477-497, (1991) · Zbl 0749.58021 [15] Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. amer. math. soc., 204, 113-135, (1975) · Zbl 0276.58005 [16] Guckeneimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983), Springer-Verlag Berlin/New York [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] 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 [21] Poincaré, H., LES Méthodes nouvelles de la Mécanique Céleste, (1897-1899), Gauthier-Villars Paris [22] P. H. Rabinowitz, Homoclinics for a singular Hamiltonian system, Geometric Analysis and the Calculus of Variations, International Press, New York [23] 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 [24] Rabinowitz, P.H., A variational approach to heteroclinic orbits for a class of Hamiltonian systems, () · Zbl 0737.58024 [25] Rabinowitz, P.H., Homoclinics for an almost periodically forced Hamiltonian system, Top. methods nonlinear anal., 6, 49-66, (1995) · Zbl 0857.34049 [26] Rabinowitz, P.H., Multibump solutions for an almost periodically forced singular Hamiltonian system, Electronic J. differential equations, 1995, (1995) · Zbl 0828.34034 [27] Rabinowitz, P.H., Heteroclinics for a reversible Hamiltonian system, Ergodic theory dynam. systems, 14, 817-829, (1994) · Zbl 0818.34025 [28] Rabinowitz, P.H., Heteroclinics for a reversible Hamiltonian system, 2, Differential integral equations, 7, 1557-1572, (1994) · Zbl 0835.34050 [29] Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27-42, (1992) · Zbl 0725.58017 [30] Séré, E., Looking for the Bernoulli shift, Ann. inst. H. Poincaré, anal. non linéaire, 10, 561-590, (1993) · Zbl 0803.58013 [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
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.