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The shadowing chain lemma for singular Hamiltonian systems involving strong forces. (English) Zbl 1269.37015

Summary: We consider a planar autonomous Hamiltonian system \(\ddot q+\nabla V(q) = 0\), where the potential \(V: \mathbb R^{2} \{\zeta \}\rightarrow \mathbb R\) has a single well of infinite depth at some point \(\zeta \) and a strict global maximum 0 at two distinct points \(a\) and \(b\). Under a strong force condition around the singularity \(\zeta \) we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits – the shadowing chain lemma – via minimization of action integrals and using simple geometrical arguments.

MSC:

37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70H05 Hamilton’s equations
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