The shadowing chain lemma for singular Hamiltonian systems involving strong forces. (English) Zbl 1269.37015

Summary: We consider a planar autonomous Hamiltonian system \(\ddot q+\nabla V(q) = 0\), where the potential \(V: \mathbb R^{2} \{\zeta \}\rightarrow \mathbb R\) has a single well of infinite depth at some point \(\zeta \) and a strict global maximum 0 at two distinct points \(a\) and \(b\). Under a strong force condition around the singularity \(\zeta \) 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.


37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70H05 Hamilton’s equations
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[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/S0308210500023349 · Zbl 0868.34001
[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
[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/S0956792506006516 · Zbl 1160.37390
[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.1996.3230 · Zbl 0887.34044
[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)80022-5 · Zbl 0907.58014
[6] Gordon W.B., Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 1975, 204, 113-135 http://dx.doi.org/10.1090/S0002-9947-1975-0377983-1 · Zbl 0276.58005
[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.2007.03.013 · Zbl 1117.37033
[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 · Zbl 1214.37049
[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 · Zbl 0701.58023
[10] Rabinowitz P.H., Homoclinics for a singular Hamiltonian system, In: Geometric Analysis and the Calculus of Variations, International Press, Cambridge, 1996, 267-296 · Zbl 0936.37035
[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_17
[12] Tanaka K., Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1990, 7(5), 427-438 · Zbl 0712.58026
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