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

Janczewska, Joanna

Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 3, pp. 471-491

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 $Q$.