The Existence and Multiplicity of Heteroclinic and Homoclinic Orbits for a Class of Singular Hamiltonian Systems in $\mathbf{R}^{2}$

Janczewska, Joanna

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

\mathbf{R}^{2}

Janczewska, Joanna

Bollettino dell'Unione Matematica Italiana, Tome 3 (2010), p. 471-491 / Harvested from Biblioteca Digitale Italiana di Matematica

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Résumé

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 $Q$ .

Publié le : 2010-10-01

MR 2742777 | Zbl 1214.37049

@article{BUMI_2010_9_3_3_471_0,
     author = {Joanna Janczewska},
     title = {The Existence and Multiplicity of Heteroclinic and Homoclinic Orbits for a Class of Singular Hamiltonian Systems in $\mathbf{R}^{2}$},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {3},
     year = {2010},
     pages = {471-491},
     zbl = {1214.37049},
     mrnumber = {2742777},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2010_9_3_3_471_0}
}

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, Tome 3 (2010) pp. 471-491. http://gdmltest.u-ga.fr/item/BUMI_2010_9_3_3_471_0/

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