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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 \[ \ddot{q}+V_{q}(t,q)=0, \tag{1} \] 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.

MSC:

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

[1] Borges M.J.: Heteroclinic and homoclinic solutions for a singular Hamiltonian system. European J. Appl. Math. 17, 1–32 (2006) · Zbl 1160.37390
[2] Caldiroli P., Jeanjean L.: Homoclinics and heteroclinics for a class of conservative singular Hamiltonian systems. J. Differential Equations 136, 76–114 (1997) · Zbl 0887.34044
[3] Gordon W.B.: Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc. 204, 113–135 (1975) · Zbl 0276.58005
[4] Izydorek M., Janczewska J.: Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differential Equations 219, 375–389 (2005) · Zbl 1080.37067
[5] Izydorek M., Janczewska J.: Heteroclinic solutions for a class of the second order Hamiltonian systems. J. Differential Equations 238, 381–393 (2007) · Zbl 1117.37033
[6] Izydorek M., Janczewska J.: The shadowing chain lemma for singular Hamiltonian systems involving strong forces. Cent. Eur. J. Math. 10, 1928–1939 (2012) · Zbl 1269.37015
[7] J. Janczewska, The existence and multiplicity of heteroclinic and homoclinic orbits for a class of singular Hamiltonian systems in $${\(\backslash\)mathbb{R}\^2}$$ . Boll. Unione Mat. Ital. (9) 3 (2010), 471–491. · Zbl 1214.37049
[8] Rabinowitz P.H.: Periodic and heteroclinic orbits for a periodic Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 6, 331–346 (1989) · Zbl 0701.58023
[9] P. H. Rabinowitz, Homoclinics for a singular Hamiltonian system. In: Geometric Analysis and the Calculus of Variations, Int. Press, Cambridge, MA, 1996, 267–296. · Zbl 0936.37035
[10] Tanaka K.: Homoclinic orbits for a singular second order Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 427–438 (1990) · Zbl 0712.58026
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