The shadowing chain lemma for singular Hamiltonian systems involving strong forces

Marek Izydorek; Joanna Janczewska

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 1928-1939
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Marek Izydorek, and Joanna Janczewska. "The shadowing chain lemma for singular Hamiltonian systems involving strong forces." Open Mathematics 10.6 (2012): 1928-1939. <http://eudml.org/doc/269571>.

@article{MarekIzydorek2012,
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. },
author = {Marek Izydorek, Joanna Janczewska},
journal = {Open Mathematics},
keywords = {Heteroclinic orbit; Homoclinic orbit; Rotation number (winding number); Shadowing chain lemma; Singular Hamiltonian systems; Strong force; heteroclinic orbit; homoclinic orbit; rotation number (winding number); shadowing chain lemma; singular Hamiltonian systems; strong force},
language = {eng},
number = {6},
pages = {1928-1939},
title = {The shadowing chain lemma for singular Hamiltonian systems involving strong forces},
url = {http://eudml.org/doc/269571},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Marek Izydorek
AU - Joanna Janczewska
TI - The shadowing chain lemma for singular Hamiltonian systems involving strong forces
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 1928
EP - 1939
AB - 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.
LA - eng
KW - Heteroclinic orbit; Homoclinic orbit; Rotation number (winding number); Shadowing chain lemma; Singular Hamiltonian systems; Strong force; heteroclinic orbit; homoclinic orbit; rotation number (winding number); shadowing chain lemma; singular Hamiltonian systems; strong force
UR - http://eudml.org/doc/269571
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., 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
  8. [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 Zbl1214.37049
  9. [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 Zbl0701.58023
  10. [10] 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
  11. [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[Crossref] 
  12. [12] 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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