Connecting orbits for a periodically forced singular planar Newtonian system. (English) Zbl 1271.34051

The authors study the existence and multiplicity of connecting orbits for a certain class of planar singular Newtonian systems \[ \ddot{q}+V_{q}(t,q)=0, \tag{1} \] i.e., solutions of (1) that emanate from a set \(\mathcal{M}\) composed of two distinct points and terminate at \(\mathcal{M}\): \(q(\pm\infty)=\lim_{t\longrightarrow\pm\infty}q(t)\in\mathcal{M}\) and \(\dot{q}(\pm\infty)=0\), with a periodic strong force \(V_{q}(t,q)\), an infinitely deep well of Gordon’s type at one point and two stationary points at which a potential \(V(t,q)\) achieves a strict global maximum. To this end, they minimize the corresponding action functional over the classes of functions in the Sobolev space \(W^{1,2}_{\mathrm{loc}}(\mathbb{R},\mathbb{R}^{2})\) that turn a given number of times around the singularity.


34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
49J27 Existence theories for problems in abstract spaces
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