The Existence and Multiplicity of Heteroclinic and Homoclinic Orbits for a Class of Singular Hamiltonian Systems in 𝐑 2

Joanna Janczewska

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 3, page 471-491
  • ISSN: 0392-4041

Abstract

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In this work we consider a class of planar second order Hamiltonian systems: q ¨ + ∇ ⁡ V ⁢ ( q ) = 0 , where a potential V has a singularity at a point ξ ∈ 𝐑 2 : V ⁢ ( q ) → - ∞ , as q → ξ and the unique global maximum 0 ∈ 𝐑 that is achieved at two distinct points a , b ∈ 𝐑 2 ∖ { ξ } . For a class of potentials that satisfy a strong force condition introduced by W. B. Gordon [Trans. Amer. Math. Soc. 204 (1975)], via minimization of action integrals, we establish the existence of at least two solutions which wind around ξ and join { a , b } to { a , b } . One of them, Q , is a heteroclinic orbit joining a to b . The second is either homoclinic or heteroclinic possessing a rotation number (a winding number) different from Q .

How to cite

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Janczewska, Joanna. "The Existence and Multiplicity of Heteroclinic and Homoclinic Orbits for a Class of Singular Hamiltonian Systems in $\mathbf{R}^{2}$." Bollettino dell'Unione Matematica Italiana 3.3 (2010): 471-491. <http://eudml.org/doc/290691>.

@article{Janczewska2010,
abstract = {In this work we consider a class of planar second order Hamiltonian systems: $\ddot\{q\} + \nabla V(q) = 0$, where a potential $V$ has a singularity at a point $\xi \in \mathbf\{R\}^\{2\}$: $V(q) \to -\infty$, as $q \to \xi$ and the unique global maximum $0 \in \mathbf\{R\}$ that is achieved at two distinct points $a,b \in \mathbf\{R\}^\{2\}\setminus \\{ \xi \\}$. For a class of potentials that satisfy a strong force condition introduced by W. B. Gordon [Trans. Amer. Math. Soc. 204 (1975)], via minimization of action integrals, we establish the existence of at least two solutions which wind around $\xi$ and join $\\{ a,b \\}$ to $\\{ a,b \\}$. One of them, $Q$, is a heteroclinic orbit joining $a$ to $b$. The second is either homoclinic or heteroclinic possessing a rotation number (a winding number) different from $Q$.},
author = {Janczewska, Joanna},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {471-491},
publisher = {Unione Matematica Italiana},
title = {The Existence and Multiplicity of Heteroclinic and Homoclinic Orbits for a Class of Singular Hamiltonian Systems in $\mathbf\{R\}^\{2\}$},
url = {http://eudml.org/doc/290691},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Janczewska, Joanna
TI - The Existence and Multiplicity of Heteroclinic and Homoclinic Orbits for a Class of Singular Hamiltonian Systems in $\mathbf{R}^{2}$
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/10//
PB - Unione Matematica Italiana
VL - 3
IS - 3
SP - 471
EP - 491
AB - In this work we consider a class of planar second order Hamiltonian systems: $\ddot{q} + \nabla V(q) = 0$, where a potential $V$ has a singularity at a point $\xi \in \mathbf{R}^{2}$: $V(q) \to -\infty$, as $q \to \xi$ and the unique global maximum $0 \in \mathbf{R}$ that is achieved at two distinct points $a,b \in \mathbf{R}^{2}\setminus \{ \xi \}$. For a class of potentials that satisfy a strong force condition introduced by W. B. Gordon [Trans. Amer. Math. Soc. 204 (1975)], via minimization of action integrals, we establish the existence of at least two solutions which wind around $\xi$ and join $\{ a,b \}$ to $\{ a,b \}$. One of them, $Q$, is a heteroclinic orbit joining $a$ to $b$. The second is either homoclinic or heteroclinic possessing a rotation number (a winding number) different from $Q$.
LA - eng
UR - http://eudml.org/doc/290691
ER -

References

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