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.


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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