Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

Joanna Janczewska; Jakub Maksymiuk

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 1920-1927
  • ISSN: 2391-5455

Abstract

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We consider a conservative second order Hamiltonian system q ¨ + ∇ V ( q ) = 0 in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ 0 = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

How to cite

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Joanna Janczewska, and Jakub Maksymiuk. "Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3." Open Mathematics 10.6 (2012): 1920-1927. <http://eudml.org/doc/269612>.

@article{JoannaJanczewska2012,
abstract = {We consider a conservative second order Hamiltonian system \[\ddot\{q\} + \nabla V(q) = 0\] in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ 0 = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.},
author = {Joanna Janczewska, Jakub Maksymiuk},
journal = {Open Mathematics},
keywords = {Homoclinic orbit; Rotation number; Strong force; homoclinic orbit; rotation number; strong force},
language = {eng},
number = {6},
pages = {1920-1927},
title = {Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3},
url = {http://eudml.org/doc/269612},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Joanna Janczewska
AU - Jakub Maksymiuk
TI - Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 1920
EP - 1927
AB - We consider a conservative second order Hamiltonian system \[\ddot{q} + \nabla V(q) = 0\] in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ 0 = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.
LA - eng
KW - Homoclinic orbit; Rotation number; Strong force; homoclinic orbit; rotation number; strong force
UR - http://eudml.org/doc/269612
ER -

References

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  1. [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[Crossref] Zbl0868.34001
  2. [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. [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[Crossref] Zbl1160.37390
  4. [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[Crossref] Zbl0887.34044
  5. [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[Crossref] Zbl0907.58014
  6. [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[Crossref] Zbl0276.58005
  7. [7] Izydorek M., Janczewska J., Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 2005, 219(2), 375–389 http://dx.doi.org/10.1016/j.jde.2005.06.029[Crossref] Zbl1080.37067
  8. [8] 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[Crossref] Zbl1117.37033
  9. [9] 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 Zbl1214.37049
  10. [10] Rabinowitz P.H., Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1989, 6(5), 331–346 Zbl0701.58023
  11. [11] Rabinowitz P.H., Homoclinics for an almost periodically forced singular Hamiltonian system, Topol. Methods Nonlinear Anal., 1995, 6(1), 49–66 Zbl0857.34049
  12. [12] Rabinowitz P.H., Multibump solutions for an almost periodically forced singular Hamiltonian system, Electron. J. Differential Equations, 1995, #12 Zbl0828.34034
  13. [13] Rabinowitz P.H., Homoclinics for a singular Hamiltonian system, In: Geometric Analysis and the Calculus of Variations, International Press, Cambridge, 1996, 267–296 Zbl0936.37035
  14. [14] Tanaka K., Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1990, 7(5), 427–438 Zbl0712.58026

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