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August  2018, 38(8): 3955-3975. doi: 10.3934/dcds.2018172

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

Carlota Rebelo 1,, and  Alexandre Simões 2,

1. 

Fac. Ciências da Univ. Lisboa e Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Campo Grande, Edifício C6, piso 2, P-1749-016 Lisboa, Portugal
2. 

Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Campo Grande, Edifício C6, piso 2, P-1749-016 Lisboa, Portugal

* Corresponding author: Carlota Rebelo

Received  October 2017 Revised  March 2018 Published  May 2018

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.

Keywords: Multiple collisions, periodic solution, Poincaré-Birkhoff theorem.
Mathematics Subject Classification: Primary: 34C25; Secondary: 37E40, 70K40.
Citation: Carlota Rebelo, Alexandre Simões. Periodic linear motions with multiple collisions in a forced Kepler type problem. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3955-3975. doi: 10.3934/dcds.2018172
References:
[1]

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.  Google Scholar

[2]

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.   Google Scholar

[3]

J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.  doi: 10.2307/1971464.  Google Scholar

[4]

M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.  Google Scholar

[5]

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.  Google Scholar

[6]

S. Marò, Periodic solution of a forced relativistic pendulum via twist dynamics, Topological Methods in Nonlinear Analysis, 42 (2013), 51-75.   Google Scholar

[7]

R. Ortega, Linear motions in a periodically forced Kepler problem, Portugaliae Mathematica, 68 (2011), 149-176.  doi: 10.4171/PM/1885.  Google Scholar

[8]

A. Simões, Bouncing solutions in a generalized Kepler problem, Master Thesis, University of Lisbon, 2016. Google Scholar

[9]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[10]

H. J. Sperling, The collision singularity in a perturbed two-body problem, Celestial Mechanics, 1 (1969/1970), 213-221.  doi: 10.1007/BF01228841.  Google Scholar

[11]

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.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

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.   Google Scholar

[3]

J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.  doi: 10.2307/1971464.  Google Scholar

[4]

M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.  Google Scholar

[5]

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.  Google Scholar

[6]

S. Marò, Periodic solution of a forced relativistic pendulum via twist dynamics, Topological Methods in Nonlinear Analysis, 42 (2013), 51-75.   Google Scholar

[7]

R. Ortega, Linear motions in a periodically forced Kepler problem, Portugaliae Mathematica, 68 (2011), 149-176.  doi: 10.4171/PM/1885.  Google Scholar

[8]

A. Simões, Bouncing solutions in a generalized Kepler problem, Master Thesis, University of Lisbon, 2016. Google Scholar

[9]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[10]

H. J. Sperling, The collision singularity in a perturbed two-body problem, Celestial Mechanics, 1 (1969/1970), 213-221.  doi: 10.1007/BF01228841.  Google Scholar

[11]

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.  Google Scholar

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