## 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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### References:

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