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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 $$\ddot{q}+V_{q}(t,q)=0, \tag1$$ 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.

34C37Homoclinic and heteroclinic solutions of ODE
49J27Optimal control problems in abstract spaces (existence)
Full Text: DOI
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