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Periodic linear motions with multiple collisions in a forced Kepler type problem

  • * Corresponding author: Carlota Rebelo

    * Corresponding author: Carlota Rebelo 
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  • In [7] the author proved the existence of multiple periodic linear motions with collisions for a periodically forced Kepler problem. We extend this result obtaining periodic solutions with multiple collisions for a forced Kepler type problem. In order to do that we apply the Poincaré-Birkhoff theorem.

    Mathematics Subject Classification: Primary: 34C25; Secondary: 37E40, 70K40.


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  • Figure 1.  The restriction of a possible (according to Proposition 2) set $D$ to the strip $0\le t_0\le 2\pi$

    Figure 2.  Cylinder $B$

  •   P. Amster , J. Haddad , R. Ortega  and  A. J. Ureña , Periodic motions in forced problems of Kepler type, NODEA, 18 (2011) , 649-657.  doi: 10.1007/s00030-011-0111-8.
      F. Dalbono  and  C. Rebelo , Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, Turin Fortnight Lectures on Nonlinear Analysis, Rend. Sem. Mat. Univ. Politec. Torino, 60 (2002) , 233-263. 
      J. Franks , Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988) , 139-151.  doi: 10.2307/1971464.
      M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.
      P. Le Calvez  and  J. Wang , Some remarks on the Poincaré-Birkhoff theorem, Proc. of the AMS, 138 (2010) , 703-715.  doi: 10.1090/S0002-9939-09-10105-3.
      S. Marò , Periodic solution of a forced relativistic pendulum via twist dynamics, Topological Methods in Nonlinear Analysis, 42 (2013) , 51-75. 
      R. Ortega , Linear motions in a periodically forced Kepler problem, Portugaliae Mathematica, 68 (2011) , 149-176.  doi: 10.4171/PM/1885.
      A. Simões, Bouncing solutions in a generalized Kepler problem, Master Thesis, University of Lisbon, 2016.
      H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.
      H. J. Sperling , The collision singularity in a perturbed two-body problem, Celestial Mechanics, 1 (1969/1970) , 213-221.  doi: 10.1007/BF01228841.
      L. Zhao , Some collision solutions of the rectilinear periodically forced Kepler problem, Adv. Nonlinear Stud., 16 (2016) , 45-49.  doi: 10.1515/ans-2015-5021.
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