Abstract
We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 \{ζ}→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits — the shadowing chain lemma — via minimization of action integrals and using simple geometrical arguments.
[1] Bertotti M.L., Jeanjean L., Multiplicity of homoclinic solutions for singular second-order conservative systems, Proc. Roy. Soc. Edinburgh Sect. A, 1996, 126(6), 1169–1180 http://dx.doi.org/10.1017/S030821050002334910.1017/S0308210500023349Search in Google Scholar
[2] Bolotin S., Variational criteria for nonintegrability and chaos in Hamiltonian systems, In: Hamiltonian Mechanics, Torun, 28 June–2 July, 1993, NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, 1994, 173–179 10.1007/978-1-4899-0964-0_14Search in Google Scholar
[3] Borges M.J., Heteroclinic and homoclinic solutions for a singular Hamiltonian system, European J. Appl. Math., 2006, 17(1), 1–32 http://dx.doi.org/10.1017/S095679250600651610.1017/S0956792506006516Search in Google Scholar
[4] Caldiroli P., Jeanjean L., Homoclinics and heteroclinics for a class of conservative singular Hamiltonian systems, J. Differential Equations, 1997, 136(1), 76–114 http://dx.doi.org/10.1006/jdeq.1996.323010.1006/jdeq.1996.3230Search in Google Scholar
[5] Caldiroli P., Nolasco M., Multiple homoclinic solutions for a class of autonomous singular systems in ℝ2, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1998, 15(1), 113–125 http://dx.doi.org/10.1016/S0294-1449(99)80022-510.1016/s0294-1449(99)80022-5Search in Google Scholar
[6] Gordon W.B., Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 1975, 204, 113–135 http://dx.doi.org/10.1090/S0002-9947-1975-0377983-110.1090/S0002-9947-1975-0377983-1Search in Google Scholar
[7] Izydorek M., Janczewska J., Heteroclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 2007, 238(2), 381–393 http://dx.doi.org/10.1016/j.jde.2007.03.01310.1016/j.jde.2007.03.013Search in Google Scholar
[8] Janczewska J., The existence and multiplicity of heteroclinic and homoclinic orbits for a class of singular Hamiltonian systems in ℝ2, Boll. Unione Mat. Ital., 2010, 3(3), 471–491 Search in Google Scholar
[9] Rabinowitz P.H., Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1989, 6(5), 331–346 10.1016/s0294-1449(16)30314-6Search in Google Scholar
[10] Rabinowitz P.H., Homoclinics for a singular Hamiltonian system, In: Geometric Analysis and the Calculus of Variations, International Press, Cambridge, 1996, 267–296 Search in Google Scholar
[11] Shil’nikov L.P., Homoclinic trajectories: from Poincaré to the present, In: Mathematical Events of the Twentieth Century, Springer, Berlin, 2006, 347–370 http://dx.doi.org/10.1007/3-540-29462-7_1710.1007/3-540-29462-7_17Search in Google Scholar
[12] Tanaka K., Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1990, 7(5), 427–438 10.1016/s0294-1449(16)30285-2Search in Google Scholar
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