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Advanced Nonlinear Studies

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Volume 16, Issue 3


Chaotic Dynamics of the Kepler Problem with Oscillating Singularity

Alessandro Margheri
  • Corresponding author
  • Faculdade de Ciências da Universidade de Lisboa e CMAF-CIO, Campo Grande, Edifício C6, piso 2, P-1749-016 Lisboa, Portugal
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/ Pedro J. Torres
Published Online: 2016-03-09 | DOI: https://doi.org/10.1515/ans-2015-5026


We prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic gravitational coefficient oscillating between two fixed values. The set of chaotic solutions we detect is coded by the number of revolutions in each period. The chaotic dynamics is obtained for large period T as well as for small angular momentum μ. In particular, we provide an explicit lower bound on T and explicit upper bound on μ which guarantee the existence of complex dynamics. We get our results by applying a simple and well-known topological method, the stretching along the path technique. Our results are robust with respect to small perturbations of the gravitational coefficient and to the addition of a small friction term.

Keywords: Gyldén Problem; Solar Radiation Pressure; Periodic Solution; Chaos; Stretching Along Paths

MSC 2010: 70F05; 34C28; 54H20


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About the article

Received: 2015-10-19

Accepted: 2015-10-23

Published Online: 2016-03-09

Published in Print: 2016-08-01

The first author was supported by Fundação para a Ciência e Tecnologia through projects UID/MAT/ 04561/2013 and PTDC/MAT/113383/2009. The second author was supported by Ministerio de Economía y Competitividad (Spain) through project MTM2014-52232-P.

Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 3, Pages 551–567, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5026.

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