Abstract
We are concerned with a planar autonomous Hamiltonian system \({\ddot{q} +\nabla V (q)=0}\), where a potential \({V : \mathbb{R}^2 \backslash \{\xi\}\rightarrow \mathbb{R}}\) has a single well of infinite depth at a point \({\xi}\) and a unique strict global maximum 0 at a point a. Under a strong force condition around the singularity \({\xi}\), via minimization of an action integral and using a shadowing chain lemma together with simple geometrical arguments, we prove the existence of infinitely many homotopy classes of \({\pi_1(\mathbb{R}^2 \backslash \{\xi\})}\) containing at least two geometrically distinct homoclinic (to a) solutions.
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Izydorek, M., Janczewska, J. Two families of infinitely many homoclinics for singular strong force Hamiltonian systems. J. Fixed Point Theory Appl. 16, 301–311 (2014). https://doi.org/10.1007/s11784-015-0212-9
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DOI: https://doi.org/10.1007/s11784-015-0212-9